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A296497 Decimal expansion of ratio-sum for A294541; see Comments.

%I A296497 #7 Jul 25 2021 19:02:12
%S A296497 3,4,8,9,8,0,1,8,3,9,1,6,5,1,8,0,8,8,3,9,3,9,5,8,4,0,8,2,8,8,2,5,7,1,
%T A296497 7,2,3,7,6,9,1,6,5,8,1,9,0,7,5,4,7,8,7,6,4,0,7,4,9,9,5,3,8,4,0,5,7,0,
%U A296497 2,8,6,8,7,8,0,0,5,7,0,3,3,7,8,9,5,3
%N A296497 Decimal expansion of ratio-sum for A294541; see Comments.
%C A296497 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 = A294541, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296497 3.489801839165180883939584082882571723769...
%t A296497 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A296497 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2];
%t A296497 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296497 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296497 Table[a[n], {n, 0, k}]; (* A294541 *)
%t A296497 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296497 Take[RealDigits[s, 10][[1]], 100]  (* A296497 *)
%Y A296497 Cf. A001622, A294541, A296284, A296498.
%K A296497 nonn,easy,cons
%O A296497 1,1
%A A296497 _Clark Kimberling_, Dec 20 2017