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A296489 Decimal expansion of ratio-sum for A293358; see Comments.

%I A296489 #7 Jul 25 2021 20:57:41
%S A296489 3,5,3,5,9,1,7,3,9,3,5,1,5,2,5,9,3,5,0,7,1,6,8,0,2,2,3,2,2,4,8,5,5,0,
%T A296489 6,7,3,3,5,5,7,5,3,4,6,2,2,7,4,4,6,8,4,2,8,0,2,2,1,0,8,9,2,8,7,4,5,0,
%U A296489 0,5,1,2,4,2,3,1,0,0,6,9,4,5,5,7,8,9
%N A296489 Decimal expansion of ratio-sum for A293358; see Comments.
%C A296489 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 = A293358, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
%e A296489 ratio-sum = 3.535917393515259350716802232248550673355...
%t A296489 a[0] = 1; a[1] = 3; b[0] = 2; b[1 ] = 4;
%t A296489 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1];
%t A296489 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296489 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296489 Table[a[n], {n, 0, k}]; (* A293358 *)
%t A296489 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296489 Take[RealDigits[s, 10][[1]], 100]  (* A296489 *)
%Y A296489 Cf. A001622, A293358, A296284, A296490.
%K A296489 nonn,easy,cons
%O A296489 1,1
%A A296489 _Clark Kimberling_, Dec 19 2017