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A296470 Decimal expansion of limiting power-ratio for A295862; see Comments.

%I A296470 #4 Dec 18 2017 19:45:57
%S A296470 6,5,1,4,7,1,0,0,7,5,0,5,5,2,7,2,7,7,3,3,5,8,8,6,8,5,7,5,3,4,3,9,6,9,
%T A296470 4,2,5,3,7,3,9,6,8,8,7,6,0,9,1,3,1,7,9,5,6,8,7,8,8,5,9,6,7,9,6,3,6,5,
%U A296470 5,1,8,0,8,3,2,6,4,8,9,8,3,8,2,4,4,0
%N A296470 Decimal expansion of limiting power-ratio for A295862; see Comments.
%C A296470 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 = A295862 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%F A296470 limiting power-ratio = 6.514710075055272773358868575343969425373...
%t A296470 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
%t A296470 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
%t A296470 j = 1; While[j < 12, k = a[j] - j - 1;
%t A296470  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296470 Table[a[n], {n, 0, 15}]  (* A295862 *)
%t A296470 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296470 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296470 Take[RealDigits[Last[h], 10][[1]], 120] (* A296470 *)
%Y A296470 Cf. A001622, A295862, A296469.
%K A296470 nonn,easy,cons
%O A296470 1,1
%A A296470 _Clark Kimberling_, Dec 18 2017