A296431 Decimal expansion of ratio-sum for A296278; see Comments.

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

Offset

2,1

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 = A296278 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 = 34.27679985968295785314965...

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 - 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}]; (* A296278 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296278 *)

Crossrefs

Cf. A001622, A296278.

Sequence in context: A166108 A255768 A216221 * A045901 A098003 A026245

Adjacent sequences: A296428 A296429 A296430 * A296432 A296433 A296434

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 14 2017

Status

approved