A136008 a(n) = n^6 - n^2.

0, 0, 60, 720, 4080, 15600, 46620, 117600, 262080, 531360, 999900, 1771440, 2985840, 4826640, 7529340, 11390400, 16776960, 24137280, 34011900, 47045520, 63999600, 85765680, 113379420, 148035360, 191102400, 244140000, 308915100

Offset: 0

Author

Rolf Pleisch, Mar 16 2008

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000290, A001014, A013661, A072691, A090986.

Programs

Magma

[6*Binomial(n^2 +1, 3): n in [0..30]]; // G. C. Greubel, Feb 07 2022

Mathematica

f[n_]:=n^6-n^2; f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,60,720,4080,15600,46620},30] (* Harvey P. Dale, Dec 07 2024 *)

Sage

[n^2*(n^4-1) for n in range(0,31)] # Zerinvary Lajos, Jul 16 2008

Formula

G.f.: 60*x^2*(1 +5*x +5*x^2 +x^3)/(1-x)^7. - Alexander R. Povolotsky, Apr 01 2008

a(n) = A001014(n) - A000290(n). - Omar E. Pol, Dec 26 2008

From Amiram Eldar, Jan 12 2021: (Start)

Sum_{n>=2} 1/a(n) = 7/8 - Pi^2/6 + Pi*coth(Pi)/4.

Sum_{n>=2} (-1)^n/a(n) = -7/8 + Pi^2/12 + Pi*csch(Pi)/4 = -7/8 + A072691 + (1/4) * A090986. (End)

a(n) = 7*a(n-1) -21*a(n-2) +35*a(n-3) -35*a(n-4) +21*a(n-5) -7*a(n-6) +a(n-7). - Wesley Ivan Hurt, May 04 2021

From G. C. Greubel, Feb 07 2022: (Start)

a(n) = 6*binomial(n^2 + 1, 3).

E.g.f.: x^2*(30 +90*x +65*x^2 +15*x^3 +x^4)*exp(x). (End)

Extensions

Extended by Ray Chandler, Dec 13 2008