A377858 - OEIS

%I #34 Nov 11 2024 09:01:44

%S 0,9,90,371,1044,2365,4654,8295,13736,21489,32130,46299,64700,88101,

%T 117334,153295,196944,249305,311466,384579,469860,568589,682110,

%U 811831,959224,1125825,1313234,1523115,1757196,2017269,2305190,2622879,2972320,3355561,3774714

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

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

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

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

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

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

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

%t LinearRecurrence[{5,-10,10,-5,1},{0,9,90,371,1044},35] (* _James C. McMahon_, Nov 10 2024 *)

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

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

%o (Python)

%o def A377858(n): return n*(n*(n*(n+2<<3)+4)-1)//3 # _Chai Wah Wu_, Nov 10 2024

%Y Cf. A014105, A376777.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Nov 09 2024