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A258783 Numerators of a rational sequence related to the closed form evaluation of a Gaussian integer zeta sum. 1
1, 1, 2, 1, 2, 2, 4, 223, 854, 4762, 1444, 2324506, 35548, 140343676, 21047728264, 88824427, 160465442, 96633020222386, 18457536052, 1397584483920886, 885721299987868, 2758893844640044, 3793843972393624, 56271391915038457502, 480348904858674456484, 85734822774933179463764, 2140418040120050844508958552 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Jonathan Borwein, Also Eisenstein
FORMULA
p(1) = 1/5, p(n) = (3/((4n+1)*(2n-3)))*Sum_{k=1..n-1} p(k)*p(n-k).
The closed form of the Gaussian integer zeta sum in question is zeta_G(4n) = p(n)*L^(4n)/(4n-1), where L is the lemniscate constant.
EXAMPLE
Sequence of fractions begins:
1/5, 1/75, 2/4875, 1/82875, 2/6215625, 2/242409375, 4/19527421875, 223/ 44815433203125, ...
MATHEMATICA
p[1] = 1/5; p[n_] := p[n] = (3/((4n+1)*(2n-3)))*Sum[p[k]*p[n-k], {k, 1, n-1}]; Table[p[n] // Numerator, {n, 1, 30}]
CROSSREFS
Cf. A062539 (lemniscate constant), A258784 (denominators).
Sequence in context: A089408 A350287 A208888 * A079318 A050315 A128978
KEYWORD
nonn,frac,easy
AUTHOR
STATUS
approved

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Last modified May 21 19:35 EDT 2024. Contains 372738 sequences. (Running on oeis4.)