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A296477 Decimal expansion of ratio-sum for A295950; see Comments.

%I A296477 #7 Jul 25 2021 20:57:22
%S A296477 4,2,8,9,9,6,9,2,7,3,6,3,8,3,1,8,1,9,5,4,8,3,3,7,9,9,6,0,2,5,5,5,3,4,
%T A296477 9,1,7,7,5,3,3,7,3,8,4,0,6,3,1,3,9,4,4,0,3,3,8,1,3,6,5,3,4,1,0,3,3,8,
%U A296477 5,4,0,8,0,0,4,2,2,2,2,9,7,0,9,7,2,5
%N A296477 Decimal expansion of ratio-sum for A295950; see Comments.
%C A296477 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 = A295950, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296477 ratio-sum = 4.289969273638318195483379960255534917753...
%t A296477 a[0] = 1; a[1] = 4; b[0] = 2; b[1 ] = 3; b[2] = 5;
%t A296477 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
%t A296477 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296477 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296477 Table[a[n], {n, 0, k}]; (* A295950 *)
%t A296477 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296477 Take[RealDigits[s, 10][[1]], 100]  (* A296477 *)
%Y A296477 Cf. A001622, A296477, A296284, A296478.
%K A296477 nonn,easy,cons
%O A296477 1,1
%A A296477 _Clark Kimberling_, Jan 05 2018