A296434 Decimal expansion of ratio-sum for A296292; see Comments.

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

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 = A296292 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=1..86.

Example

Ratio-sum = 8.012968903095661472515541...

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] + n*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}]; (* A296292 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296434 *)

Crossrefs

Cf. A001622, A296292.

Sequence in context: A307224 A309595 A329074 * A164790 A362121 A250219

Adjacent sequences: A296431 A296432 A296433 * A296435 A296436 A296437

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 15 2017

Status

approved