%I #12 Apr 29 2023 00:07:23

%S 1,0,6,2,7,5,1,6,9,9,6,9,0,2,1,1,0,7,8,2,4,5,8,3,2,5,1,9,3,3,2,6,2,6,

%T 6,9,8,2,2,7,9,5,4,2,1,1,5,1,7,2,6,6,3,1,5,7,7,2,4,0,8,4,2,6,8,1,7,1,

%U 9,1,0,5,7,9,2,3,9,1,8,7,8,5,9,0,4,0,0,9,5,8,2,1,1,2,2,3,5,7,7,1,3,8,8,8,2

%N Decimal expansion of lim_{n->oo} B(2n,7)/(B(2n)*49^n) (see comment for B(n,k) definition).

%C B(n,p) = Sum_{i=0..n} p^i * Sum_{j=0..i} binomial(n,j)*B(j) where B(k) = k-th Bernoulli number. B(2n,p)/B(2n) takes integer values for all n if p=1,2,3,4,6. p=5 is the smallest integer for which B(2n,5)/B(2n) is not always integer-valued. And lim_{n->oo} B(2n,5)/(B(2n)*25^n) = (21-sqrt(5))/16.

%F Limit_{n->oo} B(2n, 7)/(B(2n)*49^n) = 1.0627516996902110782... is the smallest root of 1728*X^3 - 6192*X^2 + 7368*X - 2911 = 0.

%t RealDigits[x/.FindRoot[ 1728x^3-6192x^2+7368x-2911==0,{x,1}, WorkingPrecision-> 120]][[1]] (* _Harvey P. Dale_, Feb 19 2012 *)

%o (PARI) solve(q=1,1.1,1728*q^3-6192*q^2+7368*q-2911)

%Y Cf. A096045, A096046, A096047, A096048, A096049.

%K cons,nonn

%O 1,3

%A _Benoit Cloitre_, Jun 17 2004