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A296469 Decimal expansion of ratio-sum for A295862; see Comments. 44
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, 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, 5, 4, 5, 4, 0, 6, 7, 3, 8, 5, 3, 9, 0, 5, 8, 8, 6, 2 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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:

****

Sequence A    ratio-sum for A     limiting power-ratio for A

A295862           A296469                 A296470

A295947           A296471                 A296472

A295948           A296473                 A296474

A295949           A296475                 A296476

A295950           A296477                 A296478

A295951           A296479                 A296480

A295952           A296481                 A296482

A295953           A296483                 A296848

A295960           A296485                 A296486

A293076           A296487                 A296488

A293358           A296489                 A296490

A294170           A296491                 A296492

A296555           A296493                 A296494

A294414           A296495                 A296496

A294541           A296497                 A296498

A294546           A296499                 A296500

A294552           A296501                 A296494

A296776           A298171                 A298172

A294553           A296503                 A296504

A296556           A296565                 A296566

A296557           A296567                 A296568

A296558           A296569                 A296570

LINKS

Table of n, a(n) for n=1..86.

EXAMPLE

ratio-sum = 6.21032710946618494227967...

MATHEMATICA

a[0] = 1; a[1] = 3; b[0] = 2; b[1 ] = 4; b[2] = 5;

a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n];

j = 1; While[j < 13, k = a[j] - j - 1;

While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];

Table[a[n], {n, 0, k}]; (* A295862 *)

g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]

Take[RealDigits[s, 10][[1]], 100]  (* A296469 *)

CROSSREFS

Cf. A001622, A296284, A296470.

Sequence in context: A306933 A316565 A011413 * A010629 A016671 A010472

Adjacent sequences:  A296466 A296467 A296468 * A296470 A296471 A296472

KEYWORD

nonn,easy,cons

AUTHOR

Clark Kimberling, Dec 18 2017

STATUS

approved

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Last modified March 28 20:01 EDT 2020. Contains 333103 sequences. (Running on oeis4.)