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A296473 Decimal expansion of ratio-sum for A295948; see Comments.

%I A296473 #10 Jul 25 2021 22:38:07
%S A296473 1,9,1,6,9,7,8,1,0,6,7,4,3,3,0,0,9,2,8,4,3,4,3,2,3,0,5,8,9,7,4,4,1,8,
%T A296473 6,0,8,1,2,1,0,1,4,8,6,7,3,4,8,9,3,4,8,1,6,8,6,2,1,7,8,6,1,9,0,5,3,1,
%U A296473 1,3,9,4,9,7,7,9,9,8,6,6,6,0,6,0,4,0
%N A296473 Decimal expansion of ratio-sum for A295948; see Comments.
%C A296473 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 = A295948, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296473 ratio-sum = 1.916978106743300928434323058974418608121...
%t A296473 a[0] = 3; a[1] = 4; b[0] = 1; b[1 ] = 2; b[2] = 5;
%t A296473 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
%t A296473 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296473 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296473 Table[a[n], {n, 0, k}]; (* A295948 *)
%t A296473 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296473 Take[RealDigits[s, 10][[1]], 100]  (* A296473 *)
%Y A296473 Cf. A001622, A295948, A296284, A296474.
%K A296473 nonn,easy,cons
%O A296473 1,2
%A A296473 _Clark Kimberling_, Dec 19 2017