A296493 Decimal expansion of ratio-sum for A296555; see Comments.

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

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 = A296555, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.

Links

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

Example

5.204091649313251611130187115558413050194...

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 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}]; (* A296555 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296493 *)

Crossrefs

Cf. A001622, A296284, A296494, A296555.

Sequence in context: A212155 A269328 A063377 * A196821 A333419 A352454

Adjacent sequences: A296490 A296491 A296492 * A296494 A296495 A296496

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 19 2017

Status

approved