A296499 Decimal expansion of ratio-sum for A294546; see Comments.

4, 7, 3, 7, 0, 5, 7, 9, 9, 8, 7, 9, 2, 2, 1, 7, 3, 6, 5, 4, 2, 8, 3, 7, 5, 6, 2, 1, 2, 9, 5, 7, 5, 9, 3, 2, 6, 1, 5, 9, 9, 7, 6, 5, 8, 7, 3, 4, 7, 0, 8, 6, 3, 7, 1, 7, 7, 5, 2, 1, 0, 2, 1, 6, 1, 5, 6, 0, 8, 0, 6, 4, 2, 9, 2, 6, 8, 7, 9, 3, 8, 1, 7, 1, 5, 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 = A294546, 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

4.737057998792217365428375621295759326159...

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 - 1] + 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}]; (* A294549 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296499 *)

Crossrefs

Cf. A001622, A294546, A296284, A296500.

Sequence in context: A271798 A190357 A372861 * A199446 A100127 A130204

Adjacent sequences: A296496 A296497 A296498 * A296500 A296501 A296502

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 20 2017

Status

approved