A296433 Decimal expansion of ratio-sum for A296288; see Comments.

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

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 = A296288 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=1..86.

Example

Ratio-sum = 7.093883244558233282518632...

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n - 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}]; (* A296288 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296433 *)

Crossrefs

Cf. A001622, A296288.

Sequence in context: A234355 A021145 A269404 * A021589 A093444 A369522

Adjacent sequences: A296430 A296431 A296432 * A296434 A296435 A296436

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 15 2017

Status

approved