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A296850
Decimal expansion of ratio-sum for A296849; see Comments.
3
2, 8, 9, 8, 4, 3, 0, 3, 7, 3, 6, 0, 5, 1, 8, 8, 6, 7, 9, 3, 6, 4, 5, 5, 4, 0, 4, 9, 9, 7, 4, 5, 3, 1, 9, 4, 1, 5, 1, 7, 4, 5, 4, 5, 4, 5, 2, 9, 5, 3, 9, 2, 4, 7, 3, 4, 6, 9, 9, 7, 5, 0, 3, 3, 3, 6, 3, 2, 6, 9, 2, 1, 8, 1, 0, 1, 7, 7, 2, 8, 4, 2, 9, 1, 5, 0
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
Cf. A296849.
Sequence in context: A108744 A195724 A081819 * A021349 A145426 A200499
KEYWORD
nonn,easy,cons
AUTHOR
Clark Kimberling, Jan 12 2018
STATUS
approved