A296475 Decimal expansion of ratio-sum for A295949; see Comments.

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

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 = A295949, 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 = 3.975789557971073473454965326842917073144...

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];
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}]; (* A295949 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100]  (* A296475 *)

Crossrefs

Cf. A001622, A295949, A296284, A296476.

Sequence in context: A193026 A296503 A265298 * A247003 A201943 A358614

Adjacent sequences: A296472 A296473 A296474 * A296476 A296477 A296478

Keyword

nonn,easy,cons

Author

Clark Kimberling, Dec 19 2017

Status

approved