A296485 Decimal expansion of ratio-sum for A295960; see Comments.

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

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 = A295960, 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

ratio-sum = 3.556958436660903995097548494451097938728...

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2; b[1 ] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[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}]; (* A295960 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296485 *)

Crossrefs

Cf. A001622, A295960, A296284, A296486.

Sequence in context: A317587 A235647 A010616 * A055594 A276173 A197631

Adjacent sequences: A296482 A296483 A296484 * A296486 A296487 A296488

Keyword

nonn,easy,cons

Author

Clark Kimberling, Apr 12 2018

Status

approved