A292227 Numerators of partial sums of the series 1 + 2*Sum_{k >= 1} 1/(4*k^4 + 1).

1, 7, 93, 467, 19173, 1170203, 19898781, 2248887383, 65223261317, 11806034873107, 694496744821, 625756401440091, 195865032043506253, 14298321093992118279, 6019647565828140441989, 222728486906331381429243, 24277533643722234159157217, 14882189966220076173164214151

Offset: 0

Author

Wolfdieter Lang, Oct 30 2017

Keywords

• nonn
• frac
• easy

Comments

The corresponding denominators are given in A292228.

The value of the series 1 + 2*Sum_{k >= 1} 1/(4*k^4 +1) is (Pi/2)*tanh(Pi/2) given in A228048. See the Koecher reference, p. 189.

Examples

			The rationals s(n) begin: 1, 7/5, 93/65, 467/325, 19173/13325, 1170203/812825, 19898781/13818025, 2248887383/1561436825,...
s(10^5) = 1.4406595199775144260 (Maple 20 digits), to be compared with 1.4406595199775145926 (20 digits from A228048).
		

References

• Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, p. 189.

Links

• Robert Israel, Table of n, a(n) for n = 0..340

Crossrefs

Cf. A228048, A292228.

Programs

Maple

seq(numer(t),t=ListTools:-PartialSums([1, seq(2/(4*k^4+1),k=1..30)]));

Mathematica

{1}~Join~Numerator[1 + 2 Accumulate[Array[1/(4 #^4 + 1) &, 17]]] (* Michael De Vlieger, Oct 30 2017 *)

PARI

a(n) = numerator(1+2*sum(k=1, n, 1/(4*k^4 + 1))); \\ Michel Marcus, Oct 30 2017

Formula

a(n) = numerators(s(n)) with the rationals (in lowest terms) s(n) = 1 + 2*Sum_{k=1..n} 1/(4*k^4 + 1), n >= 0.