A339237 Decimal expansion of K = Sum_{m>=0} 1/(1 + 2*m + 4*m^2).

1, 2, 7, 9, 7, 2, 8, 7, 4, 2, 2, 8, 1, 8, 9, 6, 8, 3, 3, 6, 4, 7, 2, 7, 5, 7, 0, 1, 5, 0, 7, 6, 3, 0, 6, 7, 2, 2, 6, 2, 6, 0, 3, 6, 7, 5, 0, 7, 5, 7, 8, 2, 6, 1, 9, 3, 0, 6, 8, 3, 0, 5, 8, 8, 1, 6, 9, 3, 0, 6, 6, 0, 7, 2, 2, 1, 3, 6, 4, 9, 0, 6, 6, 2, 1, 1, 5, 3, 2, 9, 9, 0, 5, 3, 5, 3, 2, 2, 7, 3, 7, 1, 9, 7, 1, 3, 2, 9, 2, 3

Offset: 1

Author

Artur Jasinski, Nov 28 2020

Keywords

• nonn
• cons

Comments

This constant K and the constant J = A339135 allow the expression of the real and imaginary parts of:

Psi(1/4 + i*sqrt(3)/4) = - J - log(2)/3 - (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*sqrt(3)*K;

Psi(-1/4 + i*sqrt(3)/4) = 1 - J - log(2)/3 + (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*(sqrt(3) - sqrt(3)*K + Pi*tanh(Pi*sqrt(3)/2));

Psi(3/4 + i*sqrt(3)/4)= - J - i*sqrt(3)*k - log(2)/3 + (1/2)*Pi/cosh(Pi*sqrt(3)/4) + i*Pi*tanh(Pi*sqrt(3)/2).

Psi(-3/4 + i*sqrt(3)/4) = 1 - J - log(2)/3 - (1/2)*Pi/cosh(Pi*sqrt(3)/2) + i*(sqrt(3)/3 + sqrt(3)*K).

where Psi is the digamma function and i=sqrt(-1).

Examples

			1.27972874228189683364727570150763067226260...
		

Crossrefs

Cf. A054569 (terms), A339135.

Programs

Maple

K:= Re(sum(1/(1+2*n+4*n^2), n=0..infinity)):
evalf(K, 120);  # Alois P. Heinz, Dec 06 2020

Mathematica

RealDigits[N[Re[Sum[1/(1 + 2*n + 4*n^2), {n, 0, Infinity}]], 110]][[1]]

PARI

sumpos(n=0, 1/(1+2*n+4*n^2)) \\ Michel Marcus, Nov 28 2020

Formula

Equals -i/(2*sqrt(3)) * (Psi(1/4 + i*sqrt(3)/4) - Psi(1/4 - i*sqrt(3)/4)).

Equals Pi*sqrt(3)*tanh(Pi*sqrt(3)/2)/3 - Sum_{m>=0} 1/(3 + 6*m + 4*m^2).