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%I #7 Jan 04 2018 21:23:26
%S 8,6,4,3,4,8,6,8,8,6,5,5,1,7,4,8,8,3,1,6,8,2,9,8,8,6,6,6,2,5,1,4,3,6,
%T 4,4,6,9,7,9,9,2,5,0,7,9,3,2,1,0,3,4,1,3,3,5,6,0,9,9,5,5,9,6,5,1,8,1,
%U 8,3,9,1,8,0,0,4,6,8,8,6,1,7,9,8,7,5
%N Decimal expansion of limiting power-ratio for A296555; 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 = A296555, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e limiting power-ratio = 8.643486886551748831682988666251436446979...
%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] + n;
%t j = 1; While[j < 13, 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, k}]; (* A296555 *)
%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] (* A296494 *)
%Y Cf. A001622, A296284, A296493, A296555.
%K nonn,easy,cons
%O 1,1
%A _Clark Kimberling_, Dec 19 2017
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