A297012 Decimal expansion of ratio-sum for A297011; see Comments.

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

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 = A297011, we have g = 1 + sqrt(2). 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 = 2.518439346012623900896837641191550691016...

Mathematica

a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4;
a[n_] := a[n] = 2 a[n - 1] + a[n - 2] - b[n];
j = 1; While[j < 9, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
u = Table[a[n], {n, 0, k}]; (* A297011 *)
r = 1 + Sqrt[2]; s = N[Sum[r - a[n]/a[n - 1], {n, 1, 1000}], 200];
StringJoin[StringTake[ToString[s], 41], "..."]
Take[RealDigits[s, 10][[1]], 100] (* A297012 *)

Crossrefs

Cf. A297011.

Sequence in context: A340198 A362151 A268980 * A259449 A174815 A021401

Adjacent sequences: A297009 A297010 A297011 * A297013 A297014 A297015

Keyword

nonn,easy,cons

Author

Clark Kimberling, Jan 13 2018

Status

approved