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A296478 Decimal expansion of limiting power-ratio for A295950; see Comments.

%I A296478 #4 Jan 05 2018 21:58:51
%S A296478 6,9,2,0,2,0,8,3,1,1,6,9,3,1,5,9,1,0,8,5,8,8,2,1,3,4,0,8,4,9,4,1,9,8,
%T A296478 7,0,5,7,3,8,6,7,8,6,9,1,5,5,2,6,8,4,6,2,5,3,7,6,6,9,9,3,8,5,4,7,1,8,
%U A296478 2,0,5,8,1,2,0,8,6,0,7,4,0,3,5,6,5,8
%N A296478 Decimal expansion of limiting power-ratio for A295950; see Comments.
%C A296478 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 = A295950, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296478 limiting power-ratio = 6.920208311693159108588213408494198705738...
%t A296478 a[0] = 1; a[1] = 4; b[0] = 2; b[1 ] = 3; b[2] = 5;
%t A296478 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
%t A296478 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296478  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296478 Table[a[n], {n, 0, k}]; (* A295950 *)
%t A296478 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296478 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296478 Take[RealDigits[Last[h], 10][[1]], 120]   (* A296478 *)
%Y A296478 Cf. A001622, A295950, A296284, A296477.
%K A296478 nonn,easy,cons
%O A296478 1,1
%A A296478 _Clark Kimberling_, Jan 05 2018