A008963 -id:A008963 - cp's OEIS frontend

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

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

A105503 Numbers n such that 3 is the leading digit of the n-th Fibonacci number in decimal representation.

Original entry on oeis.org

4, 9, 14, 28, 33, 38, 52, 57, 71, 76, 81, 95, 100, 105, 119, 124, 138, 143, 148, 162, 167, 172, 181, 186, 191, 205, 210, 215, 229, 234, 239, 248, 253, 258, 272, 277, 282, 296, 301, 306, 315, 320, 325, 339, 344, 349, 363, 368, 382, 387, 392, 406, 411, 416, 430

Offset: 1

Reinhard Zumkeller, Apr 11 2005

A008963(a(n)) = 3; A105513(a(n)) = A105513(a(n) - 1) + 1.

			a(10)=76: A008963(76) = A000030(A000045(76)) =
A000030(3416454622906707) = 3.

Cf. A000030, A000045, A072683, A105501, A105502, A105504, A105505, A105506, A105507, A105508, A105509.

Select[Table[{n,Fibonacci[n]},{n,450}],First[IntegerDigits[#[[2]]]]==3&][[All,1]] (* Harvey P. Dale, Apr 13 2019 *)

is(n)=digits(fibonacci(n))[1]==3 \\ Charles R Greathouse IV, Oct 07 2016

a(n) ~ kn by the equidistribution theorem, where k = log(10)/(log(4) - log(3)) = 8.00392.... - Charles R Greathouse IV, Oct 07 2016

Definition clarified by Harvey P. Dale, Apr 13 2019

A105504 Numbers m such that 4 is the leading digit of the n-th Fibonacci number in decimal representation.

Original entry on oeis.org

19, 24, 43, 48, 62, 67, 72, 86, 91, 110, 115, 129, 134, 153, 158, 177, 182, 196, 201, 220, 225, 244, 249, 263, 268, 287, 292, 311, 316, 330, 335, 354, 359, 373, 378, 383, 397, 402, 421, 426, 440, 445, 450, 464, 469, 488, 493, 507, 512, 517, 531, 536, 555, 560

Offset: 1

Reinhard Zumkeller, Apr 11 2005

A008963(a(n)) = 4; A105514(a(n)) = A105514(a(n) - 1) + 1.

			a(10)=110: A008963(110) = A000030(A000045(110)) =
A000030(43566776258854844738105) = 4.

Cf. A000030, A000045, A072702, A105501, A105502, A105503, A105505, A105506, A105507, A105508, A105509.

is(n)=digits(fibonacci(n))[1]==4 \\ Charles R Greathouse IV, Oct 07 2016

a(n) ~ kn by the equidistribution theorem, where k = log(10)/(log(5) - log(4)) = 10.318851.... - Charles R Greathouse IV, Oct 07 2016

A105506 Numbers m such that 6 is the leading digit of the n-th Fibonacci number in decimal representation.

Original entry on oeis.org

15, 20, 39, 63, 82, 87, 106, 130, 149, 154, 173, 197, 216, 221, 240, 259, 264, 283, 288, 307, 326, 331, 350, 355, 374, 393, 398, 417, 422, 441, 460, 465, 484, 508, 527, 532, 551, 575, 594, 599, 618, 642, 661, 666, 685, 709, 728, 733, 752, 771, 776, 795, 800

Offset: 1

Reinhard Zumkeller, Apr 11 2005

A008963(a(n)) = 6; A105516(a(n)) = A105516(a(n) - 1) + 1.

			a(10)=154: A008963(154) = A000030(A000045(154)) =
A000030(68330027629092351019822533679447) = 6.

Cf. A000030, A000045, A072708, A105501, A105502, A105503, A105504, A105505, A105507, A105508, A105509.

is(n)=digits(fibonacci(n))[1]==6 \\ Charles R Greathouse IV, Oct 07 2016

a(n) ~ kn by the equidistribution theorem, where k = log(10)/(log(7) - log(6)) = 14.9372.... - Charles R Greathouse IV, Oct 07 2016

A105507 Numbers m such that 7 is the leading digit of the n-th Fibonacci number in decimal representation.

Original entry on oeis.org

25, 44, 49, 68, 92, 111, 116, 135, 159, 178, 183, 202, 226, 245, 250, 269, 293, 312, 317, 336, 360, 379, 384, 403, 427, 446, 470, 489, 494, 513, 537, 556, 561, 580, 604, 623, 628, 647, 671, 690, 695, 714, 738, 757, 762, 781, 805, 824, 829, 848, 872, 891, 915

Offset: 1

Reinhard Zumkeller, Apr 11 2005

A008963(a(n)) = 7; A105517(a(n)) = A105517(a(n) - 1) + 1.

			a(10)=178: A008963(178) = A000030(A000045(178)) =
A000030(7084593923980518516849609894969925639) = 7.

Cf. A000030, A000045, A072709, A105501, A105502, A105503, A105504, A105505, A105506, A105508, A105509.

Flatten[Position[Fibonacci[Range[1000]],?(IntegerDigits[#][[1]]==7&)]] (* _Harvey P. Dale, Nov 20 2015 *)

is(n)=digits(fibonacci(n))[1]==7 \\ Charles R Greathouse IV, Oct 07 2016

a(n) ~ kn by the equidistribution theorem, where k = log(10)/(log(8) - log(7)) = 17.24377.... - Charles R Greathouse IV, Oct 07 2016

A105508 Numbers m such that 8 is the leading digit of the m-th Fibonacci number in decimal representation.

Original entry on oeis.org

6, 11, 30, 54, 73, 78, 97, 121, 140, 145, 164, 188, 207, 231, 255, 274, 298, 322, 341, 365, 389, 408, 432, 451, 456, 475, 499, 518, 523, 542, 566, 585, 590, 609, 633, 652, 676, 700, 719, 743, 767, 786, 810, 834, 853, 877, 896, 901, 920, 944, 963, 968, 987

Offset: 1

Reinhard Zumkeller, Apr 11 2005

			a(1)=6 since the 6th Fibonacci: 8 begins with 8.
a(2)=11 since the 11th Fibonacci: 89 begins with 8.

Cf. A000030, A000045, A072710, A105501, A105502, A105503, A105504, A105505, A105506, A105507, A105509.

Flatten[Position[Fibonacci[Range[1000]],?(First[IntegerDigits[#]]==8&)]] (* _Harvey P. Dale, Jan 04 2015  *)

isok(n) = digits(fibonacci(n))[1] == 8; \\ Michel Marcus, Jan 10 2014

A008963(a(n)) = A000030(A000045(a(n))) = 8.
A105518(a(n)) = A105518(a(n) - 1) + 1.
A000045(a(n)) = A045732(n).
a(n) ~ kn by the equidistribution theorem, where k = log(10)/(log(9) - log(8)) = 19.549378.... - Charles R Greathouse IV, Oct 07 2016

Example and formulas edited by Michel Marcus, Jan 10 2014

A105509 Numbers m such that 9 is the leading digit of the m-th Fibonacci number in decimal representation.

Original entry on oeis.org

16, 35, 59, 83, 102, 126, 150, 169, 193, 212, 236, 260, 279, 303, 327, 346, 370, 394, 413, 437, 461, 480, 504, 528, 547, 571, 595, 614, 638, 657, 681, 705, 724, 748, 772, 791, 815, 839, 858, 882, 906, 925, 949, 973, 992, 1016, 1040, 1059, 1083, 1102, 1107

Offset: 1

Reinhard Zumkeller, Apr 11 2005

A008963(a(n)) = 9; A105519(a(n)) = A105519(a(n) - 1) + 1.

Comment from Jonathan Vos Post, Dec 23 2006: Peterson says: "Calculate 100/89 = 1.1235955056... This fraction generates the first five Fibonacci numbers before blurring into other digits. ... 10000/9899 = 1.0102030508132134559046368... generates the first 10 Fibonacci numbers (using two digits per number). 1000000/998999 generates the first 15 Fibonacci numbers (using three digits per number). ... in successive fractions, two 0's are appended to the numerator and a 9 to the beginning and end of the denominator...."

			a(10)=21: A008963(212) = A000030(A000045(212)) =
A000030(90343046356137747723758225621187571439538669) = 9.

M. Bicknell-Johnson, A generalized magic trick from Fibonacci: Designer decimals, College Mathematics Journal 35(March):125-126, 2004.
O-Y. Chan and J. Smoak, More designer decimals: The integers and their geometric extensions College Mathematics Journal 37(November):355-363, 2006.
Ivars Peterson, Designer Decimals, Science News, Week of Nov 04 2006; Vol. 170, No. 19.
J. Smoak, and T.J. Osler, A magic trick from Fibonacci. College Mathematics Journal, 34 (2003):58-60.

Cf. A000030, A000045, A072711, A105501, A105502, A105503, A105504, A105505, A105506, A105507, A105508.

Select[Range@ 1200, First@ IntegerDigits@ Fibonacci@ # == 9 &] (* Michael De Vlieger, Aug 21 2016 *)

is(n)=digits(fibonacci(n))[1]==9 \\ Charles R Greathouse IV, Oct 07 2016

m such that d(m+5)-d(m) = 2 for d(m) = floor(1 + log_10(F(m))) and F(m) = m-th Fibonacci number = A000045(m). - Jonathan Vos Post, Dec 23 2006

a(n) ~ k*n by the equidistribution theorem, where k = 1/(1 - log(9)/log(10)) = 21.8543.... - Charles R Greathouse IV, Oct 07 2016

A138844 Concatenation of initial and final digits of n-th positive Fibonacci number.

Original entry on oeis.org

11, 11, 22, 33, 55, 88, 13, 21, 34, 55, 89, 14, 23, 37, 60, 97, 17, 24, 41, 65, 16, 11, 27, 48, 75, 13, 18, 31, 59, 80, 19, 29, 38, 57, 95, 12, 27, 39, 66, 15, 11, 26, 47, 73, 10, 13, 23, 46, 79, 15, 24, 39, 53, 82, 15, 27, 32, 59, 91, 10, 21, 41, 62, 13, 15, 28, 43, 71, 14

Offset: 1

Omar E. Pol, Apr 02 2008

Also, concatenation of A008963(n) and A003893(n).

			a(15) = 60 because the 15th positive Fibonacci number is 610 and the concatenation of initial and final digits of 610 is 60.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Benford's law

Cf. A000045, A003893, A008963, A073729, A138840, A138841, A138842, A138843.

a:= n-> (f-> parse(cat(f[1], f[-1])))(""||(combinat[fibonacci](n))):
seq(a(n), n=1..92);  # Alois P. Heinz, Nov 23 2023

FromDigits[Join[{IntegerDigits[#][[1]]},{IntegerDigits[#][[-1]]}]]&/@ Fibonacci[Range[70]] (* Harvey P. Dale, Jun 15 2018 *)

cp's OEIS Frontend

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

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A105503 Numbers n such that 3 is the leading digit of the n-th Fibonacci number in decimal representation.

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A105504 Numbers m such that 4 is the leading digit of the n-th Fibonacci number in decimal representation.

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A105506 Numbers m such that 6 is the leading digit of the n-th Fibonacci number in decimal representation.

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A105507 Numbers m such that 7 is the leading digit of the n-th Fibonacci number in decimal representation.

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A105508 Numbers m such that 8 is the leading digit of the m-th Fibonacci number in decimal representation.

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