A008963 -id:A008963

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A003893

a(n) = Fibonacci(n) mod 10.

+10
33

0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1, 0, 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

All blocks of 60 successive terms contain 20 even and 40 odd numbers. - Reinhard Zumkeller, Apr 09 2005

These are the analogs of the Fibonacci numbers in carryless arithmetic mod 10.

REFERENCES

G. Marsaglia, The mathematics of random number generators, pp. 73-90 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.

H. S. M. Coxeter, The Golden Section, Phyllotaxis and Wythoff's Game, Scripta Math. 19 (1953), 135-143. [Annotated scanned copy]

Gregory P. Dresden, Three transcendental numbers from the last non-zero digits of n^n, F_n and n!, Math. Mag., pp. 96-105, vol. 81, 2008.

Ron Knott, The Mathematical Magic of the Fibonacci Numbers

Index entries for sequences related to final digits of numbers

Index entries for sequences related to carryless arithmetic

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 1, 0, 0, -1, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1).

FORMULA

Periodic with period 60 = A001175(10).

From Reinhard Zumkeller, Apr 09 2005: (Start)

a(n) = (a(n-1) + a(n-2)) mod 10 for n > 1, a(0) = 0, a(1) = 1.

a(n) = A105471(n) - A105472(n)*10 = A105471(n)/10. (End)

a(n) = A010879(A000045(n)). - Michel Marcus, Nov 19 2022

MAPLE

with(combinat, fibonacci); A003893 := proc(n) fibonacci(n) mod 10; end;

MATHEMATICA

Table[Mod[Fibonacci[n], 10], {n, 0, 99}] (* Alonso del Arte, Jul 29 2013 *)

Table[IntegerDigits[Fibonacci[n]][[-1]], {n, 0, 99}] (* Alonso del Arte, Jul 29 2013 *)

NumberDigit[Fibonacci[Range[0, 120]], 0] (* Requires Mathematica version 12 or later *) (* Harvey P. Dale, Jul 05 2021 *)

PROG

(Haskell)

a003893 n = a003893_list !! n

a003893_list = 0 : 1 : zipWith (\u v -> (u + v) `mod` 10)

(tail a003893_list) a003893_list

-- Reinhard Zumkeller, Jul 01 2013

(PARI) a(n)=fibonacci(n)%10 \\ Charles R Greathouse IV, Feb 03 2014

(Magma) [Fibonacci(n) mod 10: n in [0..100]]; // Vincenzo Librandi, Feb 04 2014

(Python)

A003893_list, a, b, = [], 0, 1

for _ in range(10**3):

A003893_list.append(a)

a, b = b, (a+b) % 10 # Chai Wah Wu, Nov 26 2015

CROSSREFS

Cf. A000045, A010879, A089911, A105471, A105472.

Cf. A008963, A261606.

KEYWORD

nonn,base,easy

AUTHOR

N. J. A. Sloane, elipper(AT)uoft02.utoledo.edu

EXTENSIONS

More terms from Ray Chandler, Nov 15 2003

STATUS

approved

A105501

Numbers n such that 1 is the leading digit of the n-th Fibonacci number in decimal representation.

+10
11

1, 2, 7, 12, 17, 21, 22, 26, 27, 31, 36, 40, 41, 45, 46, 50, 55, 60, 64, 65, 69, 70, 74, 79, 84, 88, 89, 93, 94, 98, 103, 107, 108, 112, 113, 117, 122, 127, 131, 132, 136, 137, 141, 146, 151, 155, 156, 160, 161, 165, 170, 174, 175, 179, 180, 184, 189, 194, 198, 199

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A008963(a(n)) = 1; A105511(a(n)) = A105511(a(n) - 1) + 1.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

FORMULA

a(n) ~ kn by the equidistribution theorem, where k = log(10)/log(2) = 3.321928.... - Charles R Greathouse IV, Oct 07 2016

EXAMPLE

a(10)=31: A008963(31) = A000030(A000045(31)) =

A000030(1346269) = 1.

MAPLE

filter:= proc(n) local t;

t:= combinat:-fibonacci(n);

t < 2*10^ilog10(t)

end proc:

select(filter, [$1..200]); # Robert Israel, May 02 2018

MATHEMATICA

fQ[n_] := IntegerDigits[Fibonacci[n]][[1]] == 1; Select[Range@200, fQ] (* Robert G. Wilson v, May 02 2018 *)

PROG

(PARI) is(n)=digits(fibonacci(n))[1]==1 \\ Charles R Greathouse IV, Oct 07 2016

CROSSREFS

Cf. A000030, A000045, A072675, A105502, A105503, A105504, A105505, A105506, A105507, A105508, A105509.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Apr 11 2005

STATUS

approved

A105511

Number of times 1 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

+10
11

0, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 7, 7, 7, 7, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 12, 13, 13, 13, 13, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18, 18, 18, 19, 20, 20, 20, 20, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Winston de Greef, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = #{k: A008963(k) = 1 and 0<=k<=n};

a(A105501(n)) = a(A105501(n) - 1) + 1;

n = a(n) + A105512(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).

a(n) ~ log_10(2) * n. - Amiram Eldar, Jan 12 2023

MATHEMATICA

Accumulate[Table[If[IntegerDigits[Fibonacci[n]][[1]] == 1, 1, 0], {n, 0, 100}]] (* Amiram Eldar, Jan 12 2023 *)

PROG

(PARI)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));

(isok(n) = leadingdigit(fibonacci(n))==1);

(lista(n)=my(a=vector(1+n), r=0); for (i=1, n, r+=isok(i); a[1+i]=r); a) \\ Winston de Greef, Mar 17 2023

CROSSREFS

Cf. A000030, A000045, A008963, A105501.

Cf. A105512, A105513, A105514, A105515, A105516, A105517, A105518, A105519.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Apr 11 2005

STATUS

approved

A105519

Number of times 9 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

+10
11

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,36

LINKS

Winston de Greef, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = #{k: A008963(k) = 9 and 0<=k<=n};

a(A105509(n)) = a(A105509(n) - 1) + 1;

n = A105511(n) + A105512(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + a(n).

a(n) ~ (1 - log_10(9)) * n. - Amiram Eldar, Jan 12 2023

MATHEMATICA

Table[If[First[IntegerDigits[Fibonacci[n]]]==9, 1, 0], {n, 0, 110}]// Accumulate (* Harvey P. Dale, Nov 27 2018 *)

PROG

(PARI)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));

(isok(n) = leadingdigit(fibonacci(n))==9);

(lista(n)=my(a=vector(1+n), r=0); for (i=1, n, r+=isok(i); a[1+i]=r); a) \\ Winston de Greef, Mar 18 2023

CROSSREFS

Cf. A000030, A000045, A008963, A105509.

Cf. A105511, A105512, A105513, A105514, A105515, A105516, A105517, A105518.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Apr 11 2005

STATUS

approved

A105502

Numbers m such that 2 is the leading digit of the m-th Fibonacci number in decimal representation.

+10
10

3, 8, 13, 18, 23, 32, 37, 42, 47, 51, 56, 61, 66, 75, 80, 85, 90, 99, 104, 109, 114, 118, 123, 128, 133, 142, 147, 152, 157, 166, 171, 176, 185, 190, 195, 200, 209, 214, 219, 224, 233, 238, 243, 252, 257, 262, 267, 276, 281, 286, 291, 295, 300, 305, 310, 319

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A008963(a(n)) = 2; A105512(a(n)) = A105512(a(n) - 1) + 1.

LINKS

Winston de Greef, Table of n, a(n) for n = 1..10000

FORMULA

a(n) ~ kn by the equidistribution theorem, where k = log(10)/(log(3) - log(2)) = 5.67887.... - Charles R Greathouse IV, Oct 07 2016

EXAMPLE

a(10)=51: A008963(51) = A000030(A000045(51)) = A000030(20365011074) = 2.

MATHEMATICA

Select[Range[400], First[IntegerDigits[Fibonacci[#]]]==2&] (* Harvey P. Dale, Jul 13 2015 *)

PROG

(PARI) is(n)=digits(fibonacci(n))[1]==2 \\ Charles R Greathouse IV, Oct 07 2016

CROSSREFS

Cf. A000030, A000045, A072682, A105501, A105503, A105504, A105505, A105506, A105507, A105508, A105509.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Apr 11 2005

STATUS

approved

A105505

Numbers n such that 5 is the leading digit of the n-th Fibonacci number in decimal representation.

+10
10

5, 10, 29, 34, 53, 58, 77, 96, 101, 120, 125, 139, 144, 163, 168, 187, 192, 206, 211, 230, 235, 254, 273, 278, 297, 302, 321, 340, 345, 364, 369, 388, 407, 412, 431, 436, 455, 474, 479, 498, 503, 522, 541, 546, 565, 570, 584, 589, 608, 613, 632, 637, 651, 656

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

A008963(a(n)) = 5; A105515(a(n)) = A105515(a(n) - 1) + 1.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

FORMULA

a(n) ~ kn by the equidistribution theorem, where k = log(10)/(log(6) - log(5)) = 12.629253.... - Charles R Greathouse IV, Oct 07 2016

EXAMPLE

a(10)=120: A008963(120) = A000030(A000045(120)) =

A000030(5358359254990966640871840) = 5.

MAPLE

ld:= x -> floor(x/10^ilog10(x)):

select(n -> ld(combinat:-fibonacci(n))=5, [$1..1000]); # Robert Israel, Oct 26 2020

MATHEMATICA

Select[Range[700], First[IntegerDigits[Fibonacci[#]]]==5&] (* Harvey P. Dale, Jul 31 2018 *)

PROG

(PARI) is(n)=digits(fibonacci(n))[1]==5 \\ Charles R Greathouse IV, Oct 07 2016

CROSSREFS

Cf. A000030, A000045, A072703, A105501, A105502, A105503, A105504, A105506, A105507, A105508, A105509.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Apr 11 2005

STATUS

approved

A105512

Number of times 2 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

+10
10

0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,9

LINKS

Winston de Greef, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = #{k: A008963(k) = 2 and 0<=k<=n};

a(A105502(n)) = a(A105502(n) - 1) + 1;

n = A105511(n) + a(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).

a(n) ~ log_10(3/2) * n. - Amiram Eldar, Jan 12 2023

MATHEMATICA

Accumulate[Table[If[IntegerDigits[Fibonacci[n]][[1]] == 2, 1, 0], {n, 0, 100}]] (* Amiram Eldar, Jan 12 2023 *)

PROG

(PARI)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));

(isok(n) = leadingdigit(fibonacci(n))==2);

(lista(n)=my(a=vector(1+n), r=0); for (i=1, n, r+=isok(i); a[1+i]=r); a) \\ Winston de Greef, Mar 17 2023

CROSSREFS

Cf. A000030, A000045, A008963, A105502.

Cf. A105511, A105513, A105514, A105515, A105516, A105517, A105518, A105519.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Apr 11 2005

STATUS

approved

A105513

Number of times 3 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

+10
10

0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,10

LINKS

Winston de Greef, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = #{k: A008963(k) = 3 and 0<=k<=n};

a(A105503(n)) = a(A105503(n) - 1) + 1;

n = A105511(n) + A105512(n) + a(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).

a(n) ~ log_10(4/3) * n. - Amiram Eldar, Jan 12 2023

MATHEMATICA

Accumulate[Table[If[IntegerDigits[Fibonacci[n]][[1]] == 3, 1, 0], {n, 0, 100}]] (* Amiram Eldar, Jan 12 2023 *)

PROG

(PARI)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));

(isok(n) = leadingdigit(fibonacci(n))==3);

(lista(n)=my(a=vector(1+n), r=0); for (i=1, n, r+=isok(i); a[1+i]=r); a) \\ Winston de Greef, Mar 17 2023

CROSSREFS

Cf. A000030, A000045, A008963, A105503.

Cf. A105511, A105512, A105514, A105515, A105516, A105517, A105518, A105519.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Apr 11 2005

STATUS

approved

A105514

Number of times 4 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

+10
10

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,25

LINKS

Winston de Greef, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = #{k: A008963(k) = 4 and 0<=k<=n};

a(A105504(n)) = a(A105504(n) - 1) + 1;

n = A105511(n) + A105512(n) + A105513(n) + a(n) + A105515(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).

a(n) ~ log_10(5/4) * n. - Amiram Eldar, Jan 12 2023

MATHEMATICA

Accumulate[Table[If[IntegerDigits[Fibonacci[n]][[1]] == 4, 1, 0], {n, 0, 100}]] (* Amiram Eldar, Jan 12 2023 *)

PROG

(PARI)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));

(isok(n) = leadingdigit(fibonacci(n))==4);

(lista(n)=my(a=vector(1+n), r=0); for (i=1, n, r+=isok(i); a[1+i]=r); a) \\ Winston de Greef, Mar 17 2023

CROSSREFS

Cf. A000030, A000045, A008963, A105504.

Cf. A105511, A105512, A105513, A105515, A105516, A105517, A105518, A105519.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Apr 11 2005

STATUS

approved

A105515

Number of times 5 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

+10
10

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,11

LINKS

Winston de Greef, Table of n, a(n) for n = 0..10000

FORMULA

a(n) = #{k: A008963(k) = 5 and 0<=k<=n};

a(A105505(n)) = a(A105505(n) - 1) + 1;

n = A105511(n) + A105512(n) + A105513(n) + A105514(n) + a(n) + A105516(n) + A105517(n) + A105518(n) + A105519(n).

a(n) ~ log_10(6/5) * n. - Amiram Eldar, Jan 12 2023

MATHEMATICA

Accumulate[If[First[IntegerDigits[#]]==5, 1, 0]&/@Fibonacci[Range[0, 110]]] (* Harvey P. Dale, Nov 02 2014 *)

PROG

(PARI)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));

(isok(n) = leadingdigit(fibonacci(n))==5);

(lista(n)=my(a=vector(1+n), r=0); for (i=1, n, r+=isok(i); a[1+i]=r); a) \\ Winston de Greef, Mar 17 2023

CROSSREFS

Cf. A000030, A000045, A008963, A105505.

Cf. A105511, A105512, A105513, A105514, A105516, A105517, A105518, A105519.

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Apr 11 2005

STATUS

approved

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