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%I #7 Dec 18 2017 11:41:39
%S 3,5,9,2,9,5,4,9,2,5,5,5,8,3,1,8,4,0,9,0,2,1,6,6,6,8,7,8,3,5,1,2,1,9,
%T 1,3,2,0,7,1,5,1,8,3,9,7,5,7,9,0,8,5,6,0,7,0,8,3,0,3,1,7,9,1,0,5,2,3,
%U 9,2,8,0,5,5,2,9,5,3,9,2,1,7,7,5,4,6
%N Decimal expansion of limiting power-ratio for A296245; see Comments.
%C 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 = A296245 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 Limiting power-ratio = 35.92954925558318409021666878351219132071...
%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n]^2;
%t j = 1; While[j < 12, k = a[j] - j - 1;
%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t Table[a[n], {n, 0, 15}] (* A296245 *)
%t z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t Take[RealDigits[Last[h], 10][[1]], 120] (* A296452 *)
%Y Cf. A001622, A296245.
%K nonn,easy,cons
%O 2,1
%A _Clark Kimberling_, Dec 15 2017
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