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 = A295952, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See the guide at A296469 for related sequences.
EXAMPLE
ratio-sum = 4.845853683514362071350020567372501789034...
MATHEMATICA
a[0] = 1; a[1] = 5; b[0] = 2; b[1 ] = 3; b[2] = 4;
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}]; (* A295952 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100] (* A296481 *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Jan 05 2018
STATUS
approved