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A296492 Decimal expansion of limiting power-ratio for A294170; see Comments.

%I A296492 #9 Feb 16 2018 13:31:47
%S A296492 1,1,2,2,0,7,1,2,9,4,7,8,7,2,0,1,9,1,3,1,3,5,6,3,9,9,3,2,1,2,0,7,4,4,
%T A296492 8,2,2,3,5,2,3,0,1,4,9,2,6,1,9,0,4,2,5,0,7,7,3,3,5,9,0,7,6,1,3,8,9,6,
%U A296492 1,1,3,4,2,2,3,5,4,8,8,0,1,0,7,9,7,0
%N A296492 Decimal expansion of limiting power-ratio for A294170; see Comments.
%C A296492 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 = A294170, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296492 limiting power-ratio = 11.22071294787201913135639932120744822352...
%t A296492 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A296492 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 2 n;
%t A296492 j = 1; While[j < 16, k = a[j] - j - 1;
%t A296492 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296492 u = Table[a[n], {n, 0, k}];  (* A294170 *)
%t A296492 z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
%t A296492 StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
%t A296492 Take[RealDigits[Last[h], 10][[1]], 120]   (* A296492 *)
%Y A296492 Cf. A001622, A294381, A296284, A296491.
%K A296492 nonn,easy,cons
%O A296492 2,3
%A A296492 _Clark Kimberling_, Dec 20 2017