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A296476 Decimal expansion of limiting power-ratio for A295949; see Comments.

%I A296476 #7 Jan 07 2018 23:26:23
%S A296476 6,1,3,6,3,8,5,5,1,8,6,6,6,2,2,0,7,9,0,9,5,5,3,4,3,9,4,9,1,5,2,6,3,6,
%T A296476 1,2,4,4,6,0,1,1,9,2,3,6,9,8,1,2,8,5,1,9,0,7,9,6,7,5,7,1,7,2,0,2,5,9,
%U A296476 9,5,3,7,2,2,4,5,5,6,5,5,6,2,6,0,6,6
%N A296476 Decimal expansion of limiting power-ratio for A295949; see Comments.
%C A296476 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 = A295949, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296476 limiting power-ratio = 6.136385518666220790955343949152636124460...
%t A296476 a[0] = 1; a[1] = 2; b[0] = 3; b[1 ] = 4; b[2] = 5;
%t A296476 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
%t A296476 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296476  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296476 Table[a[n], {n, 0, k}]; (* A295949 *)
%t A296476 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296476 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296476 Take[RealDigits[Last[h], 10][[1]], 120]   (* A296476 *)
%Y A296476 Cf. A001622, A295949, A296284, A296475.
%K A296476 nonn,easy,cons
%O A296476 1,1
%A A296476 _Clark Kimberling_, Dec 19 2017
%E A296476 Definition and sequence corrected by _Clark Kimberling_, Jan 05 2018