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A298171 Decimal expansion of ratio-sum for A296776; see Comments.

%I A298171 #7 Jul 25 2021 18:20:58
%S A298171 5,6,1,1,5,6,2,0,5,7,6,5,6,2,2,5,4,7,7,0,3,2,4,4,3,4,5,6,0,9,2,5,7,9,
%T A298171 4,8,0,9,8,2,7,0,9,5,8,6,5,5,5,5,7,3,7,0,6,5,0,1,9,0,5,7,3,9,5,3,5,1,
%U A298171 0,5,4,3,3,1,7,6,6,7,6,0,2,0,1,0,5,4
%N A298171 Decimal expansion of ratio-sum for A296776; see Comments.
%C A298171 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 = A298171, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296462 for related sequences.
%e A298171 ratio-sum = 5.611562057656225477032443456092579480982...
%t A298171 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
%t A298171 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 2 n;
%t A298171 j = 1; While[j < 16, k = a[j] - j - 1;
%t A298171 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A298171 u =Table[a[n], {n, 0, k}];  (* A296776 *)
%t A298171 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A298171 Take[RealDigits[s, 10][[1]], 100]  (* A298171 *)
%Y A298171 Cf. A001622, A296462, A296776.
%K A298171 nonn,easy,cons
%O A298171 1,1
%A A298171 _Clark Kimberling_, Feb 09 2018