A105515 -id:A105515

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

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

A105516

Number of times 6 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, 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

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

OFFSET

0,21

LINKS

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

FORMULA

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

MATHEMATICA

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

PROG

(PARI)

(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

CROSSREFS

Cf. A000030, A000045, A008963, A105506.

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

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Apr 11 2005

STATUS

approved

A105517

Number of times 7 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, 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

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

OFFSET

0,45

LINKS

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

FORMULA

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

MATHEMATICA

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

PROG

(PARI)

(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

CROSSREFS

Cf. A000030, A000045, A008963, A105507.

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

KEYWORD

nonn,base

AUTHOR

Reinhard Zumkeller, Apr 11 2005

STATUS

approved

A105518

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

+10
10

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

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

OFFSET

0,12

LINKS

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

FORMULA

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

MATHEMATICA

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