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A296490 Decimal expansion of limiting power-ratio for A293358; see Comments.

%I A296490 #11 Jan 04 2018 21:23:09
%S A296490 5,7,9,2,4,9,7,4,8,4,5,2,1,0,7,5,8,9,2,7,4,7,7,3,8,0,7,5,8,8,2,8,6,7,
%T A296490 0,1,6,8,2,2,1,4,0,8,1,7,6,5,1,7,1,8,4,0,3,6,8,7,8,9,0,1,1,6,2,1,6,4,
%U A296490 9,0,0,2,9,3,3,6,0,6,8,1,1,4,4,6,9,4
%N A296490 Decimal expansion of limiting power-ratio for A293358; see Comments.
%C A296490 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 = A293358, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296490 limiting power-ratio = 5.792497484521075892747738075882867016822...
%t A296490 a[0] = 1; a[1] = 3; b[0] = 2; b[1 ] = 4;
%t A296490 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1];
%t A296490 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296490 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296490 Table[a[n], {n, 0, k}]; (* A293358 *)
%t A296490 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296490 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296490 Take[RealDigits[Last[h], 10][[1]], 120]   (* A296490 *)
%Y A296490 Cf. A001622, A293358, A296284, A296489.
%K A296490 nonn,easy,cons
%O A296490 1,1
%A A296490 _Clark Kimberling_, Dec 19 2017