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A377858
a(n) = Sum_{k=1..n} tan(k*Pi/(1+2*n))^4.
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
0,2
REFERENCES
Shigeichi Moriguchi, Kanehisa Udagawa, Shin Hitotsumatsu, "Mathematics Formulas II", Iwanami Shoten, Publishers (1957), p. 14.
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
MATHEMATICA
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 9, 90, 371, 1044}, 35] (* James C. McMahon, Nov 10 2024 *)
PROG
(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
CROSSREFS
Sequence in context: A044641 A165135 A277105 * A180289 A210088 A261315
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Nov 09 2024
STATUS
approved