A296428 Decimal expansion of ratio-sum for A295367; see Comments.

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

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 = A295367, 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

8.83944262127645153974944...

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 2]*b[n - 1];
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}]; (* A295367 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296428 *)

Crossrefs

Cf. A001622, A295367.

Sequence in context: A200686 A222066 A303617 * A073447 A011213 A178728

Adjacent sequences: A296425 A296426 A296427 * A296429 A296430 A296431

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 14 2017

Status

approved