%I #30 Feb 11 2024 23:44:58

%S 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 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 2,2,2

%N Most-significant decimal digit of Fibonacci(5n+3).

%C Leading digit of A134490(n).

%C From _Johannes W. Meijer_, Jul 06 2011: (Start)

%C 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 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 Kevin Brown, <a href="http://www.mathpages.com/home/kmath302/kmath302.htm">Benford's Law</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BenfordsLaw.html">Benford's Law</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Benford&#39;s_law">Benford's Law</a>.

%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>

%F a(n) = floor(F(5*n+3)/10^(floor(log(F(5*n+3))/log(10)))). - _Johannes W. Meijer_, Jul 06 2011

%e From _Johannes W. Meijer_, Jul 06 2011: (Start)

%e d p(N=2000) p(N=4000) p(N=6000) p(Benford)

%e 1 0.29900 0.29950 0.30033 0.30103

%e 2 0.17700 0.17675 0.17650 0.17609

%e 3 0.12550 0.12525 0.12517 0.12494

%e 4 0.09650 0.09675 0.09700 0.09691

%e 5 0.07950 0.07950 0.07933 0.07918

%e 6 0.06700 0.06675 0.06700 0.06695

%e 7 0.05800 0.05825 0.05800 0.05799

%e 8 0.05150 0.05125 0.05100 0.05115

%e 9 0.04600 0.04600 0.04567 0.04576

%e Total 1.00000 1.00000 1.00000 1.00000 (End)

%p A134490 := proc(n) combinat[fibonacci](5*n+3) ; end proc:

%p A141053 := proc(n) convert(A134490(n),base,10) ; op(-1,%) ; end proc:

%p seq(A141053(n),n=0..70) ; # _R. J. Mathar_, Jul 04 2011

%Y 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

%K nonn,base,less

%O 0,1

%A _Paul Curtz_, Aug 01 2008

%E Edited by _Johannes W. Meijer_, Jul 06 2011