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A296425
Decimal expansion of ratio-sum for A296245; see Comments.
25
1, 4, 9, 7, 6, 3, 2, 7, 1, 4, 4, 8, 5, 6, 3, 0, 4, 1, 2, 4, 1, 1, 6, 8, 9, 6, 3, 5, 6, 2, 6, 9, 8, 7, 9, 3, 6, 1, 3, 5, 1, 0, 5, 0, 4, 8, 2, 1, 7, 4, 9, 2, 0, 3, 2, 2, 3, 6, 7, 0, 3, 3, 5, 7, 8, 3, 0, 6, 8, 4, 9, 2, 4, 3, 3, 2, 4, 0, 5, 8, 2, 6, 9, 4, 7, 2
OFFSET
2,2
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 = A296245, we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.
EXAMPLE
14.9763271448563041241168963...
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]^2;
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}]; (* A296245 *)
g = GoldenRatio; s = N[Sum[- g + a[n]/a[n - 1], {n, 1, 1000}], 200]
Take[RealDigits[s, 10][[1]], 100] (* A296425 *)
CROSSREFS
Sequence in context: A245670 A166923 A021205 * A335089 A306004 A056992
KEYWORD
nonn,easy,cons
AUTHOR
Clark Kimberling, Dec 14 2017
STATUS
approved