This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A141053 #39 Sep 01 2025 18:33:03
%S A141053 2,2,2,2,2,3,3,3,4,4,5,5,6,7,8,8,9,1,1,1,1,1,1,2,2,2,2,3,3,3,4,4,5,5,
%T A141053 6,7,7,8,9,1,1,1,1,1,1,1,2,2,2,3,3,3,4,4,5,5,6,6,7,8,9,1,1,1,1,1,1,1,
%U A141053 2,2,2
%N A141053 Most-significant decimal digit of Fibonacci(5n+3).
%C A141053 Leading digit of A134490(n).
%C A141053 From _Johannes W. Meijer_, Jul 06 2011: (Start)
%C A141053 The leading digit d, 1 <= d <= 9, of A141053 follows Benford’s Law. This law states that the probability for the leading digit is p(d) = log_10(1+1/d), see the examples.
%C A141053 We observe that the last digit of A134490(n), i.e. F(5*n+3) mod 10, leads to the Lucas sequence A000032(n) (mod 10), i.e. a repetitive sequence of 12 digits [2, 1, 3, 4, 7, 1, 8, 9, 7, 6, 3, 9] with p(0) = p(5) = 0, p(1) = p(3) = p(7) = p(9) = 1/6 and p(2) = p(4) = p(6) = p(8) = 1/12. This does not obey Benford’s Law, which would predict that the last digit would satisfy p(d) = 1/10, see the links. (End)
%H A141053 Hans J. H. Tuenter, <a href="/A141053/b141053.txt">Table of n, a(n) for n = 0..10000</a>
%H A141053 Kevin Brown, <a href="http://www.mathpages.com/home/kmath302/kmath302.htm">Benford's Law</a>.
%H A141053 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BenfordsLaw.html">Benford's Law</a>.
%H A141053 Wikipedia, <a href="http://en.wikipedia.org/wiki/Benford's_law">Benford's Law</a>.
%H A141053 <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F A141053 a(n) = floor(F(5*n+3)/10^(floor(log(F(5*n+3))/log(10)))). - _Johannes W. Meijer_, Jul 06 2011
%F A141053 For n>0, a(n) = floor(10^{alpha*n+beta}), where alpha=5*log_10(phi)-1, beta=log_10(1+2/sqrt(5)), {x}=x-floor(x) denotes the fractional part of x, log_10(phi) = A097348, and phi = (1+sqrt(5))/2 = A001622. - _Hans J. H. Tuenter_, Aug 27 2025
%e A141053 From _Johannes W. Meijer_, Jul 06 2011: (Start)
%e A141053 d p(N=2000) p(N=4000) p(N=6000) p(Benford)
%e A141053 1 0.29900 0.29950 0.30033 0.30103
%e A141053 2 0.17700 0.17675 0.17650 0.17609
%e A141053 3 0.12550 0.12525 0.12517 0.12494
%e A141053 4 0.09650 0.09675 0.09700 0.09691
%e A141053 5 0.07950 0.07950 0.07933 0.07918
%e A141053 6 0.06700 0.06675 0.06700 0.06695
%e A141053 7 0.05800 0.05825 0.05800 0.05799
%e A141053 8 0.05150 0.05125 0.05100 0.05115
%e A141053 9 0.04600 0.04600 0.04567 0.04576
%e A141053 Total 1.00000 1.00000 1.00000 1.00000 (End)
%p A141053 A134490 := proc(n) combinat[fibonacci](5*n+3) ; end proc:
%p A141053 A141053 := proc(n) convert(A134490(n),base,10) ; op(-1,%) ; end proc:
%p A141053 seq(A141053(n),n=0..70) ; # _R. J. Mathar_, Jul 04 2011
%t A141053 Table[IntegerDigits[Fibonacci[5n+3]][[1]],{n,0,70}] (* _Harvey P. Dale_, Jun 22 2025 *)
%Y A141053 Cf. A000045 (F(n)), A008963 (Initial digit F(n)), A105511-A105519, A003893 (F(n) mod 10), A130893, A186190 (First digit tribonacci), A008952 (Leading digit 2^n), A008905 (Leading digit n!), A045510, A112420 (Leading digit Collatz 3*n+1 starting with 1117065), A007524 (log_10(2)), A104140 (1-log_10(9)). - _Johannes W. Meijer_, Jul 06 2011
%Y A141053 Cf. A001622, A097348.
%K A141053 nonn,base,less,changed
%O A141053 0,1
%A A141053 _Paul Curtz_, Aug 01 2008
%E A141053 Edited by _Johannes W. Meijer_, Jul 06 2011