A296495 Decimal expansion of ratio-sum for A294414; see Comments.

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

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 = A294414, 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

4.585197174184565401988869057817650746901...

Mathematica

a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1] + b[n - 2];
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}]; (* A294414 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296495 *)

Crossrefs

Cf. A001622, A294414, A296284, A296496.

Sequence in context: A016721 A089959 A085996 * A020804 A372788 A021222

Adjacent sequences: A296492 A296493 A296494 * A296496 A296497 A296498

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 20 2017

Status

approved