FIBONACCI

Xlambda · Mar 24, 2021

FIBONACCI !! recursive !! calculates the nth term of the famous Fibonacci sequence . This is more of a fun quest than a useful function. ( did this on a YT channel also, had fun with Rico about this).
This reflects how limited our computers are. You will be surprised. This time is not about recursion "iterations" limit, is about the limit of the number of calculations that a computer can perform.
Open a new workbook, define the lambda FIBONACCI. Very important, first, try values of n less then 20!! . Call FIBONACC(15) ...(19) ..etc. and increase it bit by bit.
For what number "n" do you think that your computer will start struggling giving you the answer?? Share with us here, if you please, how long it took your computer to calculate FIBONACCI(35)..(38) and (40). Eventually what type of computer you are using. Processor ,RAM etc. After all, 40 is a very small number.? Give it a try!!
Note: Fibonacci sequence is this one : 1,1,2,3,5,8,13,21,34,55,89,144....the value of n term is the sum of the previous 2 terms fib( n)=fib(n-1) + fib(n-2), and fib(1)=1,fib(2)=1
Fortunately , Fibonacci has also an explicit solution (non recursive), you can define another lambda to check the results FIB( n)=LAMBDA( n,LET(a,SQRT(5),x,((1+a)/2)^n,y,((1-a)/2)^n,1/a*(x-y)))

Excel Formula:

=LAMBDA(n,IF(n<3,1,FIBBONACCI(n-1)+FIBONACCI(n-2)))

LAMBDA 6.0.xlsx

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FIBONACCI

recursive

ckeck(non recursive)

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n value

=FIBONACCI(B3)

=FIB(B3)

time(can use phone's stopwatch)

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15

610

instant

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2584

instant

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6765

instant

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30

832040

1s

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35

9227465

?s

share this

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39088169

?s

values

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40

102334155

?s

with us ?

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FIBONACCI post

Cell Formulas
Range	Formula
C2:D2	C2	=FORMULATEXT(C3)
C3:C9	C3	=FIBONACCI(B3)
D3:D9	D3	=FIB(B3)

Xlambda · Feb 20, 2022

RRA(a) Reversed Recursive Adding !! recursive !!

Excel Formula:

=LAMBDA(a,
    LET(l,LEN(a),m,MAX(l),c,COLUMNS(a),r,SEQUENCE(m),s,SEQUENCE(,c+m-1)-r+1,
       b,MMULT(SEQUENCE(,m)^0,IFERROR((s>0)*(s<=c)*INDEX(MID(REPLACE(a,1,0,REPT(0,m-l)),r,1),m-r+1,s),0)),
       IF(AND(LEN(b)=1),FILTER(b,SEQUENCE(,COLUMNS(b))<=XMATCH(TRUE,b<>0,,-1)),RRA(b))
    )
)

The only modifications made to RA (replace one r to m-r+1)
...INDEX(MID(REPLACE(a,1,0,REPT(0,m-l)),r,1),r,s)…to ...INDEX(MID(REPLACE(a,1,0,REPT(0,m-l)),r,1),m-r+1,s)
and to last part:
instead of filtering out leading 0's we have to filter out trailing 0's
… SEQUENCE(,COLUMNS(b))>=XMATCH(TRUE,b<>0) .. to … SEQUENCE(,COLUMNS(b))<=XMATCH(TRUE,b<>0,,-1)

LARGE NUM FIBONACCI.xlsx

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Part 3. RRA. Reverse Recursive Adding Concept. Adding reversed numbers aligned to the left.

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No leading 0's are necessary and results will always be reversed.

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same example used for RA

why results will stay reversed

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=RRA(N5+8)

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reversed or flipped array

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=RRA(N11:O11+N12:O12)

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inserting 0's

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*

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notice that even from early stages using RRA will keep the

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results reversed and aligned to the left.

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this time will extract last digits as first rows,

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scnd digits to the left as scnd row+ offset to the right, etc.

see exact construction, next post

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filtering out trailing 0's

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7809 is reversed 9087

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LNF 3

Cell Formulas
Range	Formula
N8,N14	N8	=FORMULATEXT(N9)
N9:O9	N9	=RRA(N5+8)
N12	N12	=RRA(N9)
B14:E14	B14	=B11:E11+B12:E12
N15:O15	N15	=RRA(N11:O11+N12:O12)
B24:F24	B24	=B21:F21+B22:F22
B32:G32	B32	=B29:G29+B30:G30
Dynamic array formulas.

Xlambda · Feb 20, 2022

LNF( n ) Large Number Fibonacci calls RRA.

Excel Formula:

=LAMBDA(n,
    LET(f,IF({1,0},1,0),
       r,REDUCE(0,SEQUENCE( n),LAMBDA(v,i,LET(s,SEQUENCE(i),x,IFNA(INDEX(v,i-2,),0),y,INDEX(v,i-1,),z,RRA(x+y),w,IF(COLUMNS(z)=1,INDEX(z,{1,2}),z),IF(s<3,f,IF(s=i,IFERROR(w,0),v))))),
       x,INDEX(r,n,),m,COLUMNS(x),
       CONCAT(INDEX(x,SEQUENCE(,m,m,-1)))
    )
)

RRA(a)

Excel Formula:

=LAMBDA(a,
    LET(l,LEN(a),m,MAX(l),c,COLUMNS(a),r,SEQUENCE(m),s,SEQUENCE(,c+m-1)-r+1,
       b,MMULT(SEQUENCE(,m)^0,IFERROR((s>0)*(s<=c)*INDEX(MID(REPLACE(a,1,0,REPT(0,m-l)),r,1),m-r+1,s),0)),
       IF(AND(LEN(b)=1),FILTER(b,SEQUENCE(,COLUMNS(b))<=XMATCH(TRUE,b<>0,,-1)),RRA(b))
    )
)

These are the only functions we need to calculate large Fibonacci numbers.
The cost we have to pay is in calculation time. A lot of rows add up to the total number of iterations, and this is slowing down the process.
If we are patient enough, the function is capable of calculating up to F 156787 (result will have 32767 digits) without running in any other type of limitation.
This is how array construction works, followed by some large numbers calculations.

LARGE NUM FIBONACCI.xlsx

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LNF Large Number Fibonacci. The function.

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This is how LNF construction using RRA will look, first part

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B7:

=LAMBDA(n,LET(f,IF({1,0},1,0),r,REDUCE(0,SEQUENCE(n),LAMBDA(v,i,LET(s,SEQUENCE(i),x,IFERROR(INDEX(v,i-2,),0), y,INDEX(v,i-1,),z,RRA(x+y),w,IF(COLUMNS(z)=1,INDEX(z,{1,2}),z),IF(s<3,f,IF(s=i,IFERROR(w,0),v))))),r))(21)

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#N/A

LNF last part, extracting, reversing and concat n th row

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#N/A

… x,INDEX(r,n,),m,COLUMNS(x),CONCAT(INDEX(x,SEQUENCE(,m,m,-1)))))

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#N/A

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#N/A

=LET(x,INDEX(B7#,21,),m,COLUMNS(x),CONCAT(INDEX(x,SEQUENCE(,m,m,-1))))

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#N/A

10946

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#N/A

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#N/A

=LNF(21)

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#N/A

10946

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LNF 4

Cell Formulas
Range	Formula
B7:F27	B7	=LAMBDA(n,LET(f,IF({1,0},1,0),r,REDUCE(0,SEQUENCE(n),LAMBDA(v,i,LET(s,SEQUENCE(i),x,IFNA(INDEX(v,i-2,),0),y,INDEX(v,i-1,),z,RRA(x+y),w,IF(COLUMNS(z)=1,INDEX(z,{1,2}),z),IF(s<3,f,IF(s=i,IFERROR(w,0),v))))),r))(21)
H10,H13	H10	=FORMULATEXT(H11)
H11	H11	=LET(x,INDEX(B7#,21,),m,COLUMNS(x),CONCAT(INDEX(x,SEQUENCE(,m,m,-1))))
H14	H14	=LNF(21)
Dynamic array formulas.

Xlambda · Feb 20, 2022

Calculation times.
Attached, screen capture from Wolfram Alpha F 5000 .

LARGE NUM FIBONACCI.xlsx

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Large Number Fibonacci. Calculation times.

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calc time

char length

=LNF(1000)

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9.69s

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43466557686937456435688527675040625802564660517371780402481729089536555417949051890403879840079255169295922593080322634775209689623239873322471161642996440906533187938298969649928516003704476137795166849228875

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=LNF(1222)

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18.39s

256

1079969233398873236032844293301653524976362936180771647488810146258428573892502087348778468320557178381011519862132501167447528535147702705724587018435771357806213382154720957836431378302535456607039572026816018665428571946697730583021094317239872427815311

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=LNF(1477)

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32.61s

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211475298697902185255785861961179135570552502746803252174956226558634024323947666637137823932524397611864671566211908330263377425204552074188208686993669123754004340250943108709212299180422293009765404930508242975773774612140021599477983006713536106549441161323499077298115887067363710153036315849480388057657

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=LNF(2000)

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1m19.62s

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4224696333392304878706725602341482782579852840250681098010280137314308584370130707224123599639141511088446087538909603607640194711643596029271983312598737326253555802606991585915229492453904998722256795316982874482472992263901833716778060607011615497886719879858311468870876264597369086722884023654422295243347964480139515349562972087652656069529806499841977448720155612802665404554171717881930324025204312082516817125

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=LNF(5000)

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19m56.37s

1045

3878968454388325633701916308325905312082127714646245106160597214895550139044037097010822916462210669479293452858882973813483102008954982940361430156911478938364216563944106910214505634133706558656238254656700712525929903854933813928836378347518908762970712033337052923107693008518093849801803847813996748881765554653788291644268912980384613778969021502293082475666346224923071883324803280375039130352903304505842701147635242270210934637699104006714174883298422891491273104054328753298044273676822977244987749874555691907703880637046832794811358973739993110106219308149018570815397854379195305617510761053075688783766033667355445258844886241619210553457493675897849027988234351023599844663934853256411952221859563060475364645470760330902420806382584929156452876291575759142343809142302917491088984155209854432486594079793571316841692868039545309545388698114665082066862897420639323438488465240988742395873801976993820317174208932265468879364002630797780058759129671389634214252579116872755600360311370547754724604639987588046985178408674382863125

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=LAMBDA(s,c,MID(s,SEQUENCE(ROUNDUP((LEN(s)/c),0),,1,c),c))(D12,100)

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3878968454388325633701916308325905312082127714646245106160597214895550139044037097010822916462210669

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4792934528588829738134831020089549829403614301569114789383642165639441069102145056341337065586562382

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5465670071252592990385493381392883637834751890876297071203333705292310769300851809384980180384781399

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6748881765554653788291644268912980384613778969021502293082475666346224923071883324803280375039130352

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9033045058427011476352422702109346376991040067141748832984228914912731040543287532980442736768229772

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4498774987455569190770388063704683279481135897373999311010621930814901857081539785437919530561751076

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1053075688783766033667355445258844886241619210553457493675897849027988234351023599844663934853256411

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9522218595630604753646454707603309024208063825849291564528762915757591423438091423029174910889841552

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0985443248659407979357131684169286803954530954538869811466508206686289742063932343848846524098874239

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5873801976993820317174208932265468879364002630797780058759129671389634214252579116872755600360311370

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547754724604639987588046985178408674382863125

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LNF 5

Cell Formulas
Range	Formula
C4,C12,C10,C8,C6	C4	=LEN(D4)
D3,D5,D7,D9,D11,D14	D3	=FORMULATEXT(D4)
D4	D4	=LNF(1000)
D6	D6	=LNF(1222)
D8	D8	=LNF(1477)
D10	D10	=LNF(2000)
D12	D12	=LNF(5000)
D15:D25	D15	=LAMBDA(s,c,MID(s,SEQUENCE(ROUNDUP((LEN(s)/c),0),,1,c),c))(D12,100)
Dynamic array formulas.

Xlambda · Feb 21, 2022

To prove that LNF is recursive efficient, let's calculate the frequency distributions of iterations calculated by RRA for every 2 rows, maximum number of iterations needed by 2 rows, total number of iterations for entire array.
For that I will use a tool function that counts the iterations. 0 iterations means that the exit condition is met after first calculation.
I_RRA(a,[i ]) where i argument counts total number of iterations after exiting condition is met

Excel Formula:

=LAMBDA(a,[i ],
     LET(l,LEN(a),m,MAX(l),c,COLUMNS(a),r,SEQUENCE(m),s,SEQUENCE(,c+m-1)-r+1,
        b,MMULT(SEQUENCE(,m)^0,IFERROR((s>0)*(s<=c)*INDEX(MID(REPLACE(a,1,0,REPT(0,m-l)),r,1),m-r+1,s),0)),
        IF(AND(LEN(b)=1),i,I_RRA(b,i+1))
    )
)

Also, to be much easier will have ARLNF Array Reversed Large Number Fibonacci, which will return entire array, ( LNF without last part of extracting last row, reversing and concat, but without the NA()'s )
ARLNF( n)

Excel Formula:

=LAMBDA(n,
    LET(f,IF({1,0},1,0),
      r,REDUCE(0,SEQUENCE(n),LAMBDA(v,i,LET(s,SEQUENCE(i),x,IFNA(INDEX(v,i-2,),0),y,INDEX(v,i-1,),z,RRA(x+y),w,IF(COLUMNS(z)=1,INDEX(z,{1,2}),z),IF(s<3,f,IF(s=i,IFERROR(w,0),v))))),
     IFNA(r,"")
    )
)

LARGE NUM FIBONACCI.xlsx

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ARLNF Array Reversed Large Number Fibonacci

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=ARLNF(70)

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LNF 6

Cell Formulas
Range	Formula
B3	B3	=FORMULATEXT(B4)
A4:A73	A4	=SEQUENCE(70)
B4:P73	B4	=ARLNF(70)
Dynamic array formulas.

Xlambda · Feb 21, 2022

Iteration's frequency distribution for F 1000

LARGE NUM FIBONACCI.xlsx

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Entire Fibbonacci array for n=

1000

=SEQUENCE(C1)

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↓

=LET(a,F4:HF4,b,F5:HF5,y,FILTER(b,b<>""),z,INDEX(a,SEQUENCE(,COLUMNS(y))),x,IF(z="",0,z),I_RRA(x+y))

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max nr.

n

↓

=ARLNF(C1)

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iterations

1

iterations

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btwn 2 rows

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=MAX(E5:E1003)

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total nr. of

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8

0

10

iterations

7

0

3

1

11

entire array

8

0

1

2

12

=SUM(E4:E1003)

9

0

4

3

13

1289

10

0

5

14

itererations

11

0

9

8

15

frequency

12

0

4

1

16

bins

distribution

13

0

3

2

17

0

90

14

0

7

3

18

1

575

15

0

1

6

19

2

291

16

0

7

8

9

20

3

41

17

0

7

9

5

1

21

4

1

18

0

4

8

5

2

22

5

1

19

0

1

8

1

4

23

0

20

0

5

6

7

6

24

21

0

6

4

9

0

1

25

for what n we have max nr. iterations

22

0

1

7

1

26

=FILTER(D4#,ISNUMBER(XMATCH(E4:E1003,A7)))

23

0

7

5

6

8

2

27

899

24

1

8

6

3

6

4

28

25

0

5

2

0

5

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29

26

0

3

9

3

1

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1

30

27

0

8

1

4

6

9

1

31

28

0

1

8

7

1

3

32

29

0

9

2

4

1

5

33

30

0

4

0

2

3

8

34

31

0

9

6

2

6

4

3

1

35

32

0

9

0

3

8

7

1

2

36

33

1

8

7

5

4

2

5

3

37

34

0

7

8

2

0

7

5

38

35

1

5

6

4

7

2

9

39

36

0

2

5

3

0

3

9

4

1

40

37

0

7

1

8

7

5

1

4

2

41

38

0

9

6

1

8

0

9

3

42

39

1

6

8

9

5

4

2

3

6

43

40

1

5

1

4

3

2

0

1

44

41

0

1

4

1

0

8

5

6

1

45

42

0

6

9

2

4

1

9

7

6

2

46

43

1

7

3

4

9

4

3

4

47

44

1

3

7

8

0

4

1

0

7

48

45

0

7

1

3

0

9

4

3

1

49

46

0

3

0

9

1

3

6

3

8

1

50

47

0

3

7

0

5

1

2

1

7

9

2

51

48

1

6

7

9

6

2

5

7

0

8

4

52

49

1

9

4

0

2

4

7

8

7

53

50

1

5

2

0

9

6

2

6

8

5

2

1

54

51

0

4

7

0

1

0

5

6

3

0

2

55

52

0

9

0

8

2

1

5

9

2

3

56

53

0

3

7

1

9

2

6

1

3

5

57

54

0

2

7

2

1

7

5

7

6

2

6

8

58

55

0

5

4

2

6

8

3

8

5

9

3

1

59

56

0

7

1

7

3

4

1

5

8

5

2

60

57

0

2

6

1

6

9

2

5

3

4

5

6

3

LNF 7

Cell Formulas
Range	Formula
D1,E2	D1	=FORMULATEXT(D4)
F3,A26,A12,A6	F3	=FORMULATEXT(F4)
D4:D1003	D4	=SEQUENCE(C1)
F4:HF1003	F4	=ARLNF(C1)
A7	A7	=MAX(E5:E1003)
A13	A13	=SUM(E4:E1003)
A17:A22	A17	=SEQUENCE(A7+1)-1
B17:B23	B17	=FREQUENCY(E5:E1003,A17#)
A27	A27	=FILTER(D4#,ISNUMBER(XMATCH(E4:E1003,A7)))
E5:E60	E5	=LET(a,F4:HF4,b,F5:HF5,y,FILTER(b,b<>""),z,INDEX(a,SEQUENCE(,COLUMNS(y))),x,IF(z="",0,z),I_RRA(x+y))
Dynamic array formulas.

Xlambda · Feb 21, 2022

Iteration's frequency distribution for F 5000

LARGE NUM FIBONACCI.xlsx
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z AA AB
1 Entire Fibbonacci array for n= 5000 =SEQUENCE(C1)
2 ↓ =LET(a,F4:ANJ4,b,F5:ANJ5,y,FILTER(b,b<>""),z,INDEX(a,SEQUENCE(,COLUMNS(y))),x,IF(z="",0,z),I_RRA(x+y))
3 max nr. n ↓ =ARLNF(C1)
4 iterations 1 iterations 1 0
5 btwn 2 rows 2 0 1 0
6 =MAX(E5:E5003) 3 0 2 0
7 6 4 0 3 0
8 5 0 5 0
9 total nr. of 6 0 8 0
10 iterations 7 0 3 1
11 entire array 8 0 1 2
12 =SUM(E4:E5003) 9 0 4 3
13 9935 10 0 5 5
14 itererations 11 0 9 8
15 frequency frequency 12 0 4 4 1
16 bins distribution 13 0 3 3 2
17 0 90 14 0 7 7 3
18 1 1008 15 0 0 1 6
19 2 2900 16 0 7 8 9
20 3 889 17 0 7 9 5 1
21 4 101 18 0 4 8 5 2
22 5 10 19 0 1 8 1 4
23 6 1 20 0 5 6 7 6
24 0 21 0 6 4 9 0 1
25 for what n we have max nr. iterations 22 0 1 1 7 7 1
26 =FILTER(D4#,ISNUMBER(XMATCH(E4:E5003,A7))) 23 0 7 5 6 8 2
27 2201 24 1 8 6 3 6 4
28 25 0 5 2 0 5 7
29 26 0 3 9 3 1 2 1
30 27 0 8 1 4 6 9 1
31 28 0 1 1 8 7 1 3
32 29 0 9 2 2 4 1 5
33 30 0 0 4 0 2 3 8
34 31 0 9 6 2 6 4 3 1
35 32 0 9 0 3 8 7 1 2
36 33 1 8 7 5 4 2 5 3
37 34 0 7 8 8 2 0 7 5
38 35 1 5 6 4 7 2 2 9
39 36 0 2 5 3 0 3 9 4 1
40 37 0 7 1 8 7 5 1 4 2
41 38 0 9 6 1 8 8 0 9 3
42 39 1 6 8 9 5 4 2 3 6
43 40 1 5 5 1 4 3 3 2 0 1
44 41 0 1 4 1 0 8 5 5 6 1
45 42 0 6 9 2 4 1 9 7 6 2
46 43 1 7 3 4 4 9 4 3 3 4
47 44 1 3 3 7 8 0 4 1 0 7
48 45 0 0 7 1 3 0 9 4 3 1 1
49 46 0 3 0 9 1 1 3 6 3 8 1
50 47 0 3 7 0 5 1 2 1 7 9 2
51 48 1 6 7 9 6 2 5 7 0 8 4
52 49 1 9 4 0 2 4 7 8 7 7 7
53 50 1 5 2 0 9 6 2 6 8 5 2 1
54 51 0 4 7 0 1 1 0 5 6 3 0 2
55 52 0 9 9 0 0 8 2 1 5 9 2 3
56 53 0 3 7 1 1 9 2 6 1 3 3 5
57 54 0 2 7 2 1 7 5 7 6 2 6 8
58 55 0 5 4 4 2 6 8 3 8 5 9 3 1
59 56 0 7 1 7 3 3 4 1 5 8 5 2 2
60 57 0 2 6 1 6 9 2 5 3 4 5 6 3
LNF 7
Cell Formulas
Range Formula
D1,E2 D1 =FORMULATEXT(D4)
F3,A26,A12,A6 F3 =FORMULATEXT(F4)
D4:D5003 D4 =SEQUENCE(C1)
F4:ANJ5003 F4 =ARLNF(C1)
A7 A7 =MAX(E5:E5003)
A13 A13 =SUM(E4:E5003)
A17:A23 A17 =SEQUENCE(A7+1)-1
B17:B24 B17 =FREQUENCY(E5:E5003,A17#)
A27 A27 =FILTER(D4#,ISNUMBER(XMATCH(E4:E5003,A7)))
E5:E60 E5 =LET(a,F4:ANJ4,b,F5:ANJ5,y,FILTER(b,b<>""),z,INDEX(a,SEQUENCE(,COLUMNS(y))),x,IF(z="",0,z),I_RRA(x+y))
Dynamic array formulas.

Xlambda · Feb 21, 2022

LNF for n=10,000
Attached, screen capture from Wolfram Alpha F 10,000 . Fibonacci 10000 - Wolfram|Alpha

LARGE NUM FIBONACCI.xlsx

A

B

C

D

1

Large Number Fibonacci n=10,000

2

3

char length

=LNF(10000)

4

2090

33644764876431783266621612005107543310302148460680063906564769974680081442166662368155595513633734025582065332680836159373734790483865268263040892463056431887354544369559827491606602099884183933864652731300088830269235673613135117579297437854413752130520504347701602264758318906527890855154366159582987279682987510631200575428783453215515103870818298969791613127856265033195487140214287532698187962046936097879900350962302291026368131493195275630227837628441540360584402572114334961180023091208287046088923962328835461505776583271252546093591128203925285393434620904245248929403901706233888991085841065183173360437470737908552631764325733993712871937587746897479926305837065742830161637408969178426378624212835258112820516370298089332099905707920064367426202389783111470054074998459250360633560933883831923386783056136435351892133279732908133732642652633989763922723407882928177953580570993691049175470808931841056146322338217465637321248226383092103297701648054726243842374862411453093812206564914032751086643394517512161526545361333111314042436854805106765843493523836959653428071768775328348234345557366719731392746273629108210679280784718035329131176778924659089938635459327894523777674406192240337638674004021330343297496902028328145933418826817683893072003634795623117103101291953169794607632737589253530772552375943788434504067715555779056450443016640119462580972216729758615026968443146952034614932291105970676243268515992834709891284706740862008587135016260312071903172086094081298321581077282076353186624611278245537208532365305775956430072517744315051539600905168603220349163222640885248852433158051534849622434848299380905070483482449327453732624567755879089187190803662058009594743150052402532709746995318770724376825907419939632265984147498193609285223945039707165443156421328157688908058783183404917434556270520223564846495196112460268313970975069382648706613264507665074611512677522748621598642530711298441182622661057163515069260029861704945425047491378115154139941550671256271197133252763631939606902895650288268608362241082050562430701794976171121233066073310059947366875

5

6

=LAMBDA(s,c,MID(s,SEQUENCE(ROUNDUP((LEN(s)/c),0),,1,c),c))(D4,120)

7

336447648764317832666216120051075433103021484606800639065647699746800814421666623681555955136337340255820653326808361593

8

737347904838652682630408924630564318873545443695598274916066020998841839338646527313000888302692356736131351175792974378

9

544137521305205043477016022647583189065278908551543661595829872796829875106312005754287834532155151038708182989697916131

10

278562650331954871402142875326981879620469360978799003509623022910263681314931952756302278376284415403605844025721143349

11

611800230912082870460889239623288354615057765832712525460935911282039252853934346209042452489294039017062338889910858410

12

651831733604374707379085526317643257339937128719375877468974799263058370657428301616374089691784263786242128352581128205

13

163702980893320999057079200643674262023897831114700540749984592503606335609338838319233867830561364353518921332797329081

14

337326426526339897639227234078829281779535805709936910491754708089318410561463223382174656373212482263830921032977016480

15

547262438423748624114530938122065649140327510866433945175121615265453613331113140424368548051067658434935238369596534280

16

717687753283482343455573667197313927462736291082106792807847180353291311767789246590899386354593278945237776744061922403

17

376386740040213303432974969020283281459334188268176838930720036347956231171031012919531697946076327375892535307725523759

18

437884345040677155557790564504430166401194625809722167297586150269684431469520346149322911059706762432685159928347098912

19

847067408620085871350162603120719031720860940812983215810772820763531866246112782455372085323653057759564300725177443150

20

515396009051686032203491632226408852488524331580515348496224348482993809050704834824493274537326245677558790891871908036

21

620580095947431500524025327097469953187707243768259074199396322659841474981936092852239450397071654431564213281576889080

22

587831834049174345562705202235648464951961124602683139709750693826487066132645076650746115126775227486215986425307112984

23

411826226610571635150692600298617049454250474913781151541399415506712562711971332527636319396069028956502882686083622410

24

82050562430701794976171121233066073310059947366875

25

LNF 9

Cell Formulas
Range	Formula
D3,D6	D3	=FORMULATEXT(D4)
C4	C4	=LEN(D4)
D4	D4	=LNF(10000)
D7:D24	D7	=LAMBDA(s,c,MID(s,SEQUENCE(ROUNDUP((LEN(s)/c),0),,1,c),c))(D4,120)
Dynamic array formulas.

Xlambda · Feb 21, 2022

Iteration's frequency distribution for F 10,000

LARGE NUM FIBONACCI.xlsx
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z AA AB
1 Entire Fibbonacci array for n= 10000 =SEQUENCE(C1)
2 ↓ =LET(a,F4:CBO4,b,F5:CBO5,y,FILTER(b,b<>""),z,INDEX(a,SEQUENCE(,COLUMNS(y))),x,IF(z="",0,z),I_RRA(x+y))
3 max nr. n ↓ =ARLNF(C1)
4 iterations 1 iterations 1 0
5 btwn 2 rows 2 0 1 0
6 =MAX(E5:E10003) 3 0 2 0
7 6 4 0 3 0
8 5 0 5 0
9 total nr. of 6 0 8 0
10 iterations 7 0 3 1
11 entire array 8 0 1 2
12 =SUM(E4:E10003) 9 0 4 3
13 22736 10 0 5 5
14 itererations 11 0 9 8
15 frequency frequency 12 0 4 4 1
16 bins distribution 13 0 3 3 2
17 0 90 14 0 7 7 3
18 1 1017 15 0 0 1 6
19 2 5416 16 0 7 8 9
20 3 3066 17 0 7 9 5 1
21 4 365 18 0 4 8 5 2
22 5 41 19 0 1 8 1 4
23 6 4 20 0 5 6 7 6
24 0 21 0 6 4 9 0 1
25 for what n we have max nr. iterations 22 0 1 1 7 7 1
26 =FILTER(D4#,ISNUMBER(XMATCH(E4:E10003,A7))) 23 0 7 5 6 8 2
27 2201 24 1 8 6 3 6 4
28 6299 25 0 5 2 0 5 7
29 8790 26 0 3 9 3 1 2 1
30 9780 27 0 8 1 4 6 9 1
31 28 0 1 1 8 7 1 3
32 29 0 9 2 2 4 1 5
33 30 0 0 4 0 2 3 8
34 31 0 9 6 2 6 4 3 1
35 32 0 9 0 3 8 7 1 2
36 33 1 8 7 5 4 2 5 3
37 34 0 7 8 8 2 0 7 5
38 35 1 5 6 4 7 2 2 9
39 36 0 2 5 3 0 3 9 4 1
40 37 0 7 1 8 7 5 1 4 2
41 38 0 9 6 1 8 8 0 9 3
42 39 1 6 8 9 5 4 2 3 6
43 40 1 5 5 1 4 3 3 2 0 1
44 41 0 1 4 1 0 8 5 5 6 1
45 42 0 6 9 2 4 1 9 7 6 2
46 43 1 7 3 4 4 9 4 3 3 4
47 44 1 3 3 7 8 0 4 1 0 7
48 45 0 0 7 1 3 0 9 4 3 1 1
49 46 0 3 0 9 1 1 3 6 3 8 1
50 47 0 3 7 0 5 1 2 1 7 9 2
51 48 1 6 7 9 6 2 5 7 0 8 4
52 49 1 9 4 0 2 4 7 8 7 7 7
53 50 1 5 2 0 9 6 2 6 8 5 2 1
54 51 0 4 7 0 1 1 0 5 6 3 0 2
55 52 0 9 9 0 0 8 2 1 5 9 2 3
56 53 0 3 7 1 1 9 2 6 1 3 3 5
57 54 0 2 7 2 1 7 5 7 6 2 6 8
58 55 0 5 4 4 2 6 8 3 8 5 9 3 1
59 56 0 7 1 7 3 3 4 1 5 8 5 2 2
60 57 0 2 6 1 6 9 2 5 3 4 5 6 3
LNF 8
Cell Formulas
Range Formula
D1,E2 D1 =FORMULATEXT(D4)
F3,A26,A12,A6 F3 =FORMULATEXT(F4)
D4:D10003 D4 =SEQUENCE(C1)
F4:CBO10003 F4 =ARLNF(C1)
A7 A7 =MAX(E5:E10003)
A13 A13 =SUM(E4:E10003)
A17:A23 A17 =SEQUENCE(A7+1)-1
B17:B24 B17 =FREQUENCY(E5:E10003,A17#)
A27:A30 A27 =FILTER(D4#,ISNUMBER(XMATCH(E4:E10003,A7)))
E5:E60 E5 =LET(a,F4:CBO4,b,F5:CBO5,y,FILTER(b,b<>""),z,INDEX(a,SEQUENCE(,COLUMNS(y))),x,IF(z="",0,z),I_RRA(x+y))
Dynamic array formulas.

Xlambda · Feb 21, 2022

Function of "n" that estimates total number of iterations "ni" for calculating LNF(n )
ni(n )=5E-05*n^2+1.8851*n-676.31

LARGE NUM FIBONACCI.xlsx
A B C D E F G H I J K L
1 =SCAN(0,'LNF 8'!E5:E10003,LAMBDA(v,x,v+x))
2 iterations =XLOOKUP(D6#,A4#,B4#)
3 n running tot ↓ =0.00005*D6#^2+1.8851*D6#-676.31 =0.00005*H6#^2+1.8851*H6#-676.31
4 2 0 total function's function's
5 3 0 n iterations estimation n estimation
6 4 0 500 517 279 1000 1,259
7 5 0 1000 1289 1,259 2000 3,294
8 6 0 1500 2198 2,264 3000 5,429
9 7 0 2000 3168 3,294 4000 7,664
10 8 0 2500 4202 4,349 5000 9,999
11 9 0 3000 5294 5,429 6000 12,434
12 10 0 3500 6438 6,534 7000 14,969
13 11 0 4000 7567 7,664 8000 17,604
14 12 0 4500 8736 8,819 9000 20,340
15 13 0 5000 9935 9,999 10000 23,175
16 14 0 5500 11149 11,204 11000 26,110
17 15 0 6000 12388 12,434 12000 29,145
18 16 0 6500 13640 13,689 13000 32,280
19 17 0 7000 14896 14,969 14000 35,515
20 18 0 7500 16165 16,274 15000 38,850
21 19 0 8000 17472 17,604 16000 42,285
22 20 0 8500 18778 18,960 17000 45,820
23 21 0 9000 20092 20,340 18000 49,455
24 22 0 9500 21417 21,745 19000 53,191
25 23 0 10000 22736 23,175 20000 57,026
26 24 1 21000 60,961
27 25 1 22000 64,996
28 26 1 23000 69,131
29 27 1 24000 73,366
30 28 1 25000 77,701
31 29 1 26000 82,136
32 30 1 27000 86,671
33 31 1 28000 91,306
34 32 1 29000 96,042
35 33 2 30000 100,877
36 34 2 31000 105,812
37 35 3 32000 110,847
38 36 3 33000 115,982
39 37 3 34000 121,217
40 38 3 35000 126,552
41 39 4 36000 131,987
42 40 5 37000 137,522
43 41 5 38000 143,157
44 42 5 39000 148,893
45 43 6 40000 154,728
46 44 7 41000 160,663
47 45 7 42000 166,698
48 46 7 43000 172,833
49 47 7 44000 179,068
50 48 8 45000 185,403
51 49 9 46000 191,838
52 50 10 47000 198,373
53 51 10 48000 205,008
54 52 10 49000 211,744
55 53 10 50000 218,579
56 54 10 51000 225,514
57 55 10 52000 232,549
58 56 10 53000 239,684
59 57 10 54000 246,919
60 58 11 55000 254,254
61 59 12 56000 261,689
62 60 13 57000 269,224
63 61 13 58000 276,859
64 62 14 59000 284,595
65 63 15 60000 292,430
66 64 16
LNF 10
Cell Formulas
Range Formula
B1,F3,I3 B1 =FORMULATEXT(B4)
E2 E2 =FORMULATEXT(E6)
A4:A10002 A4 =SEQUENCE(10000-1)+1
B4:B10002 B4 =SCAN(0,'LNF 8'!E5:E10003,LAMBDA(v,x,v+x))
D6:D25 D6 =SEQUENCE(20,,500,500)
E6:E25 E6 =XLOOKUP(D6#,A4#,B4#)
F6:F25 F6 =0.00005*D6#^2+1.8851*D6#-676.31
H6:H65 H6 =SEQUENCE(60,,1000,1000)
I6:I65 I6 =0.00005*H6#^2+1.8851*H6#-676.31
Dynamic array formulas.

jdellasala · Dec 8, 2022

Lots of chatter here, but no mention that the LAMBDA posted has a typo. Posted is
"=LAMBDA(n,IF(n<3,1,FIBBONACCI(n-1)+FIBONACCI(n-2)))", but for any value 3 or more it just generates a #NAME! error. Should be

Excel Formula:

=LAMBDA(n,IF(n<3,1,FIBONACCI(n-1)+FIBONACCI(n-2)))

Just sayin'.

FIBONACCI

Xlambda

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Cell Formulas
Range	Formula
B1,F3,I3	B1	=FORMULATEXT(B4)
E2	E2	=FORMULATEXT(E6)
A4:A10002	A4	=SEQUENCE(10000-1)+1
B4:B10002	B4	=SCAN(0,'LNF 8'!E5:E10003,LAMBDA(v,x,v+x))
D6:D25	D6	=SEQUENCE(20,,500,500)
E6:E25	E6	=XLOOKUP(D6#,A4#,B4#)
F6:F25	F6	=0.00005D6#^2+1.8851D6#-676.31
H6:H65	H6	=SEQUENCE(60,,1000,1000)
I6:I65	I6	=0.00005H6#^2+1.8851H6#-676.31
Dynamic array formulas.

FIBONACCI

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Attachments

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