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%I #4 Jan 13 2018 09:28:39
%S 2,2,8,3,3,7,8,4,1,4,0,4,2,9,9,5,1,2,2,7,9,9,6,6,6,6,0,0,3,9,8,6,7,8,
%T 7,4,9,0,8,9,9,2,8,2,7,5,3,5,4,8,8,4,9,3,7,3,9,1,2,5,0,6,6,3,2,3,2,0,
%U 7,6,1,0,5,2,0,8,9,0,2,7,0,6,1,3,5,8
%N Decimal expansion of limiting power-ratio for A296849; 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 = A296849, we have g = 1+ sqrt(2). See the guide at A296469 for related sequences.
%e limiting power-ratio = 2.283378414042995122799666600398678749089...
%t a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t a[n_] := a[n] = 2 a[n - 1] + a[n - 2] + b[n];
%t j = 1; While[j < 8, k = a[j] - j - 1;
%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t u = Table[a[n], {n, 0, k}]; (* A296849 *)
%t z = 1700; r = 1 + Sqrt[2]; h = Table[N[a[n]/r^n, z], {n, 0, z}];
%t StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t Take[RealDigits[Last[h], 10][[1]], 120] (* A296851 *)
%Y Cf. A296849, A296850.
%K nonn,easy,cons
%O 1,1
%A _Clark Kimberling_, Jan 13 2018
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