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A296480 Decimal expansion of limiting power-ratio for A295951; see Comments.

%I A296480 #4 Jan 05 2018 21:59:08
%S A296480 6,7,4,9,9,1,8,6,6,2,0,4,9,9,8,5,4,2,4,8,2,8,6,9,9,4,6,5,3,9,4,5,6,5,
%T A296480 2,9,3,9,8,7,5,7,7,8,3,5,3,9,8,6,1,5,6,8,6,8,6,8,3,7,0,5,3,8,9,4,4,1,
%U A296480 6,6,7,1,9,7,8,9,3,8,2,8,9,1,7,9,7,4
%N A296480 Decimal expansion of limiting power-ratio for A295951; see Comments.
%C A296480 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 = A295951, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296480 limiting power-ratio = 6.749918662049985424828699465394565293987...
%t A296480 a[0] = 2; a[1] = 3; b[0] = 1; b[1 ] = 4; b[2] = 5;
%t A296480 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
%t A296480 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296480  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296480 Table[a[n], {n, 0, k}]; (* A295951 *)
%t A296480 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296480 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296480 Take[RealDigits[Last[h], 10][[1]], 120]   (* A296480 *)
%Y A296480 Cf. A001622, A295951, A296284, A296479.
%K A296480 nonn,easy,cons
%O A296480 1,1
%A A296480 _Clark Kimberling_, Jan 05 2018