A377858 a(n) = Sum_{k=1..n} tan(k*Pi/(1+2*n))^4.

0, 9, 90, 371, 1044, 2365, 4654, 8295, 13736, 21489, 32130, 46299, 64700, 88101, 117334, 153295, 196944, 249305, 311466, 384579, 469860, 568589, 682110, 811831, 959224, 1125825, 1313234, 1523115, 1757196, 2017269, 2305190, 2622879, 2972320, 3355561, 3774714

Offset: 0

Author

Seiichi Manyama, Nov 09 2024

Keywords

• nonn
• easy

References

• Shigeichi Moriguchi, Kanehisa Udagawa, Shin Hitotsumatsu, "Mathematics Formulas II", Iwanami Shoten, Publishers (1957), p. 14.

Links

• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A014105, A376777.

Programs

Mathematica

LinearRecurrence[{5,-10,10,-5,1},{0,9,90,371,1044},35] (* James C. McMahon, Nov 10 2024 *)

PARI

a(n) = n*(2*n+1)*(4*n^2+6*n-1)/3;

PARI

my(N=40, x='x+O('x^N)); concat(0, Vec(x*(9+45*x+11*x^2-x^3)/(1-x)^5))

Python

def A377858(n): return n*(n*(n*(n+2<<3)+4)-1)//3 # Chai Wah Wu, Nov 10 2024

Formula

a(n) = n * (2*n+1) * (4*n^2+6*n-1)/3.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.

G.f.: x * (9 + 45*x + 11*x^2 - x^3)/(1 - x)^5.

E.g.f.: exp(x)*x*(27 + 108*x + 64*x^2 + 8*x^3)/3. - Stefano Spezia, Nov 10 2024