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