A297012 - OEIS

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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

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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], "..."]