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%I #7 Jul 25 2021 20:58:35
%S 2,8,2,0,3,3,9,6,1,8,7,5,3,5,4,3,6,8,9,2,9,9,5,5,5,3,2,9,1,4,8,4,4,1,
%T 2,8,1,5,6,5,8,1,5,4,6,0,9,6,8,6,1,9,4,5,8,1,5,2,2,3,4,1,8,9,1,7,9,3,
%U 7,0,9,4,1,0,7,7,7,0,4,6,2,0,8,9,5,1
%N Decimal expansion of ratio-sum for A294558; see Comments.
%C 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 = A294558, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e ratio-sum = 2.820339618753543689299555329148441281565...
%t a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
%t a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2] - n;
%t j = 1; While[j < 13, k = a[j] - j - 1;
%t While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t Table[a[n], {n, 0, k}]; (* A294558 *)
%t g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t Take[RealDigits[s, 10][[1]], 100] (* A296569 *)
%Y Cf. A001622, A294558, A296469, A296570.
%K nonn,easy,cons
%O 1,1
%A _Clark Kimberling_, Jan 06 2018
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