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A296474 Decimal expansion of limiting power-ratio for A295948; see Comments.

%I A296474 #4 Dec 19 2017 18:38:53
%S A296474 7,4,3,2,1,3,8,6,5,6,0,2,2,4,6,4,6,9,8,6,0,3,7,4,4,7,6,2,9,9,9,1,5,0,
%T A296474 0,0,7,5,5,1,2,5,5,0,7,1,9,6,0,8,2,8,5,9,9,8,0,3,1,0,5,5,1,5,0,6,3,4,
%U A296474 8,4,1,8,0,3,4,0,8,6,9,6,6,3,4,8,6,3
%N A296474 Decimal expansion of limiting power-ratio for A295948; see Comments.
%C A296474 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 = A296948 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296474 limiting power-ratio = 7.432138656022464698603744762999150007551...
%t A296474 a[0] = 3; a[1] = 4; b[0] = 1; b[1 ] = 2; b[2] = 5;
%t A296474 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
%t A296474 j = 1; While[j < 12, k = a[j] - j - 1;
%t A296474  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296474 Table[a[n], {n, 0, 15}]  (* A295948 *)
%t A296474 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296474 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296474 Take[RealDigits[Last[h], 10][[1]], 120] (* A296474 *)
%Y A296474 Cf. A001622, A295948, A296473.
%K A296474 nonn,easy,cons
%O A296474 1,1
%A A296474 _Clark Kimberling_, Dec 19 2017