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A298172 Decimal expansion of limiting power-ratio for A296776; see Comments.

%I A298172 #4 Feb 09 2018 17:36:47
%S A298172 1,1,6,5,8,3,7,7,4,8,5,0,5,6,4,6,6,6,8,1,7,0,8,6,8,2,6,0,6,2,9,3,9,4,
%T A298172 9,4,7,3,9,2,0,5,5,6,7,8,1,6,7,0,6,2,8,1,8,0,6,9,4,5,7,6,0,9,0,5,4,8,
%U A298172 1,9,3,4,6,0,0,2,0,5,9,7,2,8,1,3,5,9
%N A298172 Decimal expansion of limiting power-ratio for A296776; see Comments.
%C A298172 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 = A296776, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296462 for related sequences.
%e A298172 limiting power-ratio = 11.65837748505646668170868260629394947392...
%t A298172 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
%t A298172 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 2 n;
%t A298172 j = 1; While[j < 16, k = a[j] - j - 1;
%t A298172  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A298172 u = Table[a[n], {n, 0, k}];  (* A296776 *)
%t A298172 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A298172 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A298172 Take[RealDigits[Last[h], 10][[1]], 120]   (* A298172 *)
%Y A298172 Cf. A001622, A296462, A296776.
%K A298172 nonn,easy,cons
%O A298172 2,3
%A A298172 _Clark Kimberling_, Feb 09 2018