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 = A296849, we have g = 1 + sqrt(2). See A296425..A296434 for related ratio-sums and A296452..A296461 for related limiting power-ratios.
EXAMPLE
ratio-sum = 2.898430373605188679364554049974531941517...
MATHEMATICA
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = 2*a[n - 1] + a[n - 2] + b[n];
j = 1; While[j < 8, 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}]; (* A296849 *)
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] (* A296850 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Jan 12 2018
STATUS
approved