A296432 Decimal expansion of ratio-sum for A296284; see Comments.

6, 2, 1, 0, 3, 2, 7, 1, 0, 9, 4, 6, 6, 1, 8, 4, 9, 4, 2, 2, 7, 9, 6, 7, 9, 0, 4, 8, 4, 0, 2, 4, 2, 2, 4, 6, 0, 5, 4, 5, 3, 6, 8, 4, 1, 5, 7, 0, 9, 5, 7, 9, 1, 2, 3, 4, 0, 6, 9, 2, 7, 3, 5, 8, 7, 0, 5, 4, 0, 4, 4, 9, 1, 7, 0, 1, 8, 9, 8, 8, 8, 9, 6, 2, 7, 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 = A296284 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 = 6.21032710946618494227967...

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3;
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}]; (* A296284 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296432 *)

Crossrefs

Cf. A001622, A296284.

Sequence in context: A182639 A244135 A120002 * A171542 A335587 A320302

Adjacent sequences: A296429 A296430 A296431 * A296433 A296434 A296435

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 15 2017

Status

approved