A105515 -id:A105515 - cp's OEIS frontend

A105511 Number of times 1 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105501.

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

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

(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

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

A105519 Number of times 9 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105509.

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

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

(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

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

A105505 Numbers n such that 5 is the leading digit of the n-th Fibonacci number in decimal representation.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Apr 11 2005

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

			a(10)=120: A008963(120) = A000030(A000045(120)) =
A000030(5358359254990966640871840) = 5.

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

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

ld:= x -> floor(x/10^ilog10(x)):
select(n -> ld(combinat:-fibonacci(n))=5, [$1..1000]); # Robert Israel, Oct 26 2020

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

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

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

A105512 Number of times 2 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105502.

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

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

(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

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

A105513 Number of times 3 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105503.

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

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

(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

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

A105514 Number of times 4 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105504.

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

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

(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

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

A105516 Number of times 6 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

Original entry on oeis.org

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, 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, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105506.

Cf. A105511, A105512, A105513, A105514, A105515, A105517, A105518, A105519.

Prepend[Accumulate[If[First[IntegerDigits[#]]==6,1,0]&/@Fibonacci[ Range[ 110]]],0] (* Harvey P. Dale, Feb 18 2011 *)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));
(isok(n) = leadingdigit(fibonacci(n))==6);
(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

a(n) = #{k: A008963(k) = 6 and 0<=k<=n};
a(A105506(n)) = a(A105506(n) - 1) + 1;
n = A105511(n) + A105512(n) + A105513(n) + A105514(n) + A105515(n) + a(n) + A105517(n) + A105518(n) + A105519(n).
a(n) ~ log_10(7/6) * n. - Amiram Eldar, Jan 12 2023

A105517 Number of times 7 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 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, 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, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105507.

Cf. A105511, A105512, A105513, A105514, A105515, A105516, A105518, A105519.

Accumulate[Table[If[IntegerDigits[Fibonacci[n]][[1]]==7,1,0],{n,0,120}]] (* Harvey P. Dale, Apr 29 2018 *)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));
(isok(n) = leadingdigit(fibonacci(n))==7);
(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

a(n) = #{k: A008963(k) = 7 and 0<=k<=n};
a(A105507(n)) = a(A105507(n) - 1) + 1;
n = A105511(n) + A105512(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + a(n) + A105518(n) + A105519(n).
a(n) ~ log_10(8/7) * n. - Amiram Eldar, Jan 12 2023

A105518 Number of times 8 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

Original entry on oeis.org

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, 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, 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

Offset: 0

Reinhard Zumkeller, Apr 11 2005

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

Cf. A000030, A000045, A008963, A105508.

Cf. A105511, A105512, A105513, A105514, A105515, A105516, A105517, A105519.

Accumulate[Table[If[IntegerDigits[Fibonacci[n]][[1]] == 8, 1, 0], {n, 0, 100}]] (* Amiram Eldar, Jan 12 2023 *)

(leadingdigit(n, b=10) = n \ 10^logint(n, b));
(isok(n) = leadingdigit(fibonacci(n))==8);
(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

a(n) = #{k: A008963(k) = 8 and 0<=k<=n};
a(A105508(n)) = a(A105508(n) - 1) + 1;
n = A105511(n) + A105512(n) + A105513(n) + A105514(n) + A105515(n) + A105516(n) + A105517(n) + a(n) + A105519(n).
a(n) ~ log_10(9/8) * n. - Amiram Eldar, Jan 12 2023

cp's OEIS Frontend

A105511 Number of times 1 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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A105519 Number of times 9 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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A105505 Numbers n such that 5 is the leading digit of the n-th Fibonacci number in decimal representation.

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A105512 Number of times 2 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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A105513 Number of times 3 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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A105514 Number of times 4 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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A105516 Number of times 6 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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A105517 Number of times 7 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.

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