A296483 Decimal expansion of ratio-sum for A295953; see Comments.

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

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 = A295953, 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 = 4.194678679056371758991840818123954420964...

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}]; (* A295953 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296483 *)

Crossrefs

Cf. A001622, A295953, A296284, A296848.

Sequence in context: A132020 A175643 A143864 * A073364 A125165 A259448

Adjacent sequences: A296480 A296481 A296482 * A296484 A296485 A296486

Keyword

nonn,easy,cons

Author

Clark Kimberling, Jan 06 2018

Status

approved