A363773 a(n) = (4^(n+1) + (-1)^n + 5)/10.

1, 2, 7, 26, 103, 410, 1639, 6554, 26215, 104858, 419431, 1677722, 6710887, 26843546, 107374183, 429496730, 1717986919, 6871947674, 27487790695, 109951162778, 439804651111, 1759218604442, 7036874417767, 28147497671066, 112589990684263, 450359962737050

Offset: 0

Author

Evangelos G. Filothodoros, Jun 21 2023

Keywords

• nonn
• easy

Comments

a(n) is a part of the numerator of the approximate solutions x(n) = (Pi/2)*(1+5/((4^(n+1)-(-1)^(n+1)))) = a(n)*Pi/A015521(n+1) of D_d(exp(-i*x(n))) = Cl_d(x(n)+Pi) = 0, where D_d(exp(-i*x(n))) is the Bloch-Wigner-Ramakrishnan polylogarithm function and Cl_d(x(n)+Pi) is the Clausen function for odd d >= 3 and n >= 0.

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000
• Evangelos G. Filothodoros, Anastasios C. Petkou, and Nicholas D. Vlachos, The fermion-boson map for large d, Nuclear Physics B, Volume 941, 2019, pp. 195-224.
• Index entries for linear recurrences with constant coefficients, signature (4,1,-4).

Crossrefs

Cf. A015521, A037481.

Programs

Mathematica

A363773list[nmax_]:=LinearRecurrence[{4,1,-4},{1,2,7},nmax+1];A363773list[50] (* Paolo Xausa, Jun 29 2023 *)

Python

def A363773(n): return (1<<(n<<1|1))//5+1 # Chai Wah Wu, Jun 28 2023

Formula

a(n) = 1 + A037481(n).

G.f.: (1-2*x-2*x^2)/((x-1)*(4*x-1)*(x+1)).

E.g.f.: (4*e^(4*x) + e^-x + 5*e^x)/10.