A296430 Decimal expansion of ratio-sum for A296272; see Comments.

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

Offset

2,2

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 = A296272 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.

Links

Table of n, a(n) for n=2..87.

Example

ratio-sum = 12.5831861005560957189096...

Mathematica

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n - 1]*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}]; (* A296272 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296430 *)

Crossrefs

Cf. A001622, A296272.

Sequence in context: A131598 A220337 A198545 * A220398 A352633 A200225

Adjacent sequences: A296427 A296428 A296429 * A296431 A296432 A296433

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 14 2017

Status

approved