A296471 Decimal expansion of ratio-sum for A295947; see Comments.

2, 4, 2, 7, 1, 7, 9, 4, 8, 8, 0, 5, 6, 0, 3, 9, 4, 2, 4, 4, 2, 3, 6, 5, 3, 1, 0, 3, 8, 3, 1, 4, 5, 2, 2, 5, 1, 7, 5, 7, 9, 1, 6, 7, 4, 0, 4, 7, 2, 5, 2, 8, 1, 6, 7, 7, 2, 3, 6, 8, 5, 3, 1, 6, 1, 6, 1, 1, 0, 1, 7, 9, 1, 4, 9, 8, 4, 2, 4, 8, 6, 3, 8, 9, 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 = A295947, 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 = 2.427179488056039424423653103831452251757...

Mathematica

a[0] = 2; a[1] = 4; b[0] = 1; b[1 ] = 3; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[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}]; (* A295947 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296471 *)

Crossrefs

Cf. A001622, A295947, A296284, A296472.

Sequence in context: A361727 A261964 A177847 * A021416 A094756 A307667

Adjacent sequences: A296468 A296469 A296470 * A296472 A296473 A296474

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 18 2017

Status

approved