A296501 Decimal expansion of ratio-sum for A294552; see Comments.

6, 3, 5, 8, 6, 8, 5, 3, 1, 1, 1, 1, 1, 1, 9, 2, 3, 2, 1, 0, 0, 9, 7, 0, 1, 9, 7, 0, 0, 5, 7, 1, 5, 6, 4, 8, 3, 9, 3, 9, 4, 1, 4, 1, 1, 9, 4, 3, 6, 9, 0, 9, 1, 9, 3, 2, 6, 3, 4, 6, 3, 6, 9, 2, 2, 9, 5, 5, 1, 6, 1, 6, 1, 2, 7, 7, 9, 4, 6, 7, 8, 2, 7, 3, 1, 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 = A294552, 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

6.358685311111192321009701970057156483939...

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] + b[n - 2] + 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}]; (* A294552 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296501 *)

Crossrefs

Cf. A001622, A294552, A296469.

Sequence in context: A171030 A294386 A217769 * A296491 A085653 A022462

Adjacent sequences: A296498 A296499 A296500 * A296502 A296503 A296504

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 20 2017

Status

approved