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A296469 Decimal expansion of ratio-sum for A295862; see Comments.

%I A296469 #17 Jul 25 2021 22:38:21
%S A296469 3,8,7,0,2,3,6,0,7,9,7,9,5,9,5,9,3,2,3,2,8,2,0,5,2,3,1,1,7,8,3,9,9,5,
%T A296469 0,1,3,8,5,6,7,3,9,8,3,0,0,9,7,2,3,1,9,9,4,3,0,1,0,8,7,6,5,5,9,5,8,0,
%U A296469 5,4,5,4,0,6,7,3,8,5,3,9,0,5,8,8,6,2
%N A296469 Decimal expansion of ratio-sum for A295862; see Comments.
%C A296469 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 = A295862, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.  Guide to more ratio-sums and limiting power-ratios:
%C A296469 ****
%C A296469 Sequence A    ratio-sum for A     limiting power-ratio for A
%C A296469 A295862           A296469                 A296470
%C A296469 A295947           A296471                 A296472
%C A296469 A295948           A296473                 A296474
%C A296469 A295949           A296475                 A296476
%C A296469 A295950           A296477                 A296478
%C A296469 A295951           A296479                 A296480
%C A296469 A295952           A296481                 A296482
%C A296469 A295953           A296483                 A296848
%C A296469 A295960           A296485                 A296486
%C A296469 A293076           A296487                 A296488
%C A296469 A293358           A296489                 A296490
%C A296469 A294170           A296491                 A296492
%C A296469 A296555           A296493                 A296494
%C A296469 A294414           A296495                 A296496
%C A296469 A294541           A296497                 A296498
%C A296469 A294546           A296499                 A296500
%C A296469 A294552           A296501                 A296494
%C A296469 A296776           A298171                 A298172
%C A296469 A294553           A296503                 A296504
%C A296469 A296556           A296565                 A296566
%C A296469 A296557           A296567                 A296568
%C A296469 A296558           A296569                 A296570
%e A296469 ratio-sum = 6.21032710946618494227967...
%t A296469 a[0] = 1; a[1] = 3; b[0] = 2; b[1 ] = 4; b[2] = 5;
%t A296469 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];
%t A296469 j = 1; While[j < 13, k = a[j] - j - 1;
%t A296469 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296469 Table[a[n], {n, 0, k}]; (* A295862 *)
%t A296469 g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
%t A296469 Take[RealDigits[s, 10][[1]], 100]  (* A296469 *)
%Y A296469 Cf. A001622, A296284, A296470.
%K A296469 nonn,easy,cons
%O A296469 1,1
%A A296469 _Clark Kimberling_, Dec 18 2017