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A296496 Decimal expansion of limiting power-ratio for A294414; see Comments.

%I A296496 #7 Jul 25 2021 20:57:56
%S A296496 8,8,1,4,1,0,4,6,3,2,2,0,2,5,6,5,5,2,7,9,2,5,1,8,8,3,2,2,5,8,5,4,1,2,
%T A296496 6,7,8,5,0,8,3,6,4,9,7,9,6,8,7,2,7,4,8,4,8,8,8,3,0,9,3,6,0,3,5,4,6,5,
%U A296496 5,5,7,7,8,9,9,6,6,4,4,2,8,3,9,0,5,3
%N A296496 Decimal expansion of limiting power-ratio for A294414; see Comments.
%C A296496 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 ratio-sum for A is |a(1)/a(0) - g| + |a(2)/a(1) - g| + ..., assuming that this series converges. For A = A294414, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296496 limiting power-ratio = 8.814104632202565527925188322585412678508...
%t A296496 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A296496 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2];
%t A296496 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296496 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296496 Table[a[n], {n, 0, k}]; (* A294414 *)
%t A296496 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296496 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296496 Take[RealDigits[Last[h], 10][[1]], 120]   (* A296496 *)
%Y A296496 Cf. A001622, A294414, A296284, A296495.
%K A296496 nonn,easy,cons
%O A296496 1,1
%A A296496 _Clark Kimberling_, Dec 20 2017