A296427 Decimal expansion of ratio-sum for A296257; see Comments.

7, 4, 8, 6, 5, 9, 8, 2, 3, 8, 8, 6, 1, 1, 7, 5, 1, 5, 0, 8, 3, 0, 4, 2, 2, 9, 1, 1, 8, 2, 1, 9, 2, 9, 6, 7, 7, 6, 4, 4, 6, 8, 2, 9, 9, 2, 3, 8, 9, 5, 1, 2, 3, 4, 6, 7, 0, 7, 0, 8, 2, 3, 7, 2, 0, 1, 3, 6, 8, 1, 1, 7, 6, 2, 5, 8, 9, 6, 9, 1, 6, 7, 0, 6, 1, 2

Offset

2,1

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 = A296257, 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.

Links

Table of n, a(n) for n=2..87.

Example

7.48659823886117515083042...

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2]^2;
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}]; (* A296257 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296427 *)

Crossrefs

Cf. A001622, A296257.

Sequence in context: A161166 A199060 A330596 * A092034 A153042 A330595

Adjacent sequences: A296424 A296425 A296426 * A296428 A296429 A296430

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 14 2017

Status

approved