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A296459 Decimal expansion of limiting power-ratio for A296284; see Comments.

%I A296459 #4 Dec 18 2017 19:45:21
%S A296459 1,6,7,1,5,8,3,7,9,3,0,7,8,7,2,9,5,5,9,5,9,9,0,7,0,1,3,3,3,1,5,6,5,9,
%T A296459 1,7,0,8,8,6,1,6,6,5,1,3,7,6,5,4,7,0,3,7,0,6,4,6,4,7,2,2,5,9,2,2,9,6,
%U A296459 6,0,0,5,3,8,4,1,9,5,4,8,0,4,7,0,9,4
%N A296459 Decimal expansion of limiting power-ratio for A296284; see Comments.
%C A296459 Suppose that A = (a(n)), for n >=0, is a sequence, and g is a real number such that a(n)/a(n-1) -> g. The limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A296284 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.
%e A296459 limiting power-ratio = 16.71583793078729559599070133315659170886...
%t A296459 a[0] = 1; a[1] = 2; b[0] = 3;
%t A296459 a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n - 2];
%t A296459 j = 1; While[j < 12, k = a[j] - j - 1;
%t A296459  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296459 Table[a[n], {n, 0, 15}]  (* A296284 *)
%t A296459 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296459 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296459 Take[RealDigits[Last[h], 10][[1]], 120] (* A296459 *)
%Y A296459 Cf. A001622, A296284.
%K A296459 nonn,easy,cons
%O A296459 2,2
%A A296459 _Clark Kimberling_, Dec 18 2017