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%I #4 Jan 13 2018 09:29:17
%S 9,1,9,4,0,1,9,8,6,4,2,9,2,6,9,6,7,5,8,3,1,3,2,7,0,0,0,4,6,1,7,4,2,5,
%T 9,6,8,8,7,7,7,0,5,4,9,1,9,4,8,8,1,0,8,7,9,8,9,8,9,6,9,5,9,7,5,2,6,5,
%U 0,6,9,1,2,7,1,5,3,0,5,0,6,9,7,2,5,1
%N Decimal expansion of limiting power-ratio for A297011; 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 = A297011, we have g = 1+ sqrt(2). See the guide at A296469 for related sequences.
%e limiting power-ratio = 0.919401986429269675831327000461742596887...
%t a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4;
%t a[n_] := a[n] = 2 a[n - 1] + a[n - 2] - b[n];
%t j = 1; While[j < 9, 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}]; (* A297011 *)
%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] (* A297013 *)
%Y Cf. A297011.
%K nonn,easy,cons
%O 0,1
%A _Clark Kimberling_, Jan 13 2018
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