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A296488 Decimal expansion of limiting power-ratio for A293076; see Comments.

%I A296488 #7 Jan 01 2018 21:57:28
%S A296488 4,8,6,3,6,9,8,8,6,8,1,5,6,0,7,9,1,9,5,8,5,9,8,8,8,7,5,2,1,4,9,6,5,7,
%T A296488 1,9,8,7,1,7,4,9,0,9,2,2,2,5,6,9,4,8,8,2,3,8,9,7,6,2,2,3,2,9,1,6,7,9,
%U A296488 6,4,4,5,0,1,6,1,7,1,3,3,9,0,8,6,3,9
%N A296488 Decimal expansion of limiting power-ratio for A293076; see Comments.
%C A296488 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 = A293076, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296488 limiting power-ratio = 4.863698868156079195859888752149657198717...
%t A296488 a[0] = 1; a[1] = 3; b[0] = 2; b[1 ] = 4; b[2] = 5;
%t A296488 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2];
%t A296488 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296488 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296488 Table[a[n], {n, 0, k}]; (* A293076 *)
%t A296488 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296488 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296488 Take[RealDigits[Last[h], 10][[1]], 120]   (* A296488 *)
%Y A296488 Cf. A001622, A293076, A296284, A296487.
%K A296488 nonn,easy,cons
%O A296488 1,1
%A A296488 _Clark Kimberling_, Dec 19 2017