A168566 a(n) = (n-1)*(n+2)*(n^2 + n + 2)/4.

0, 8, 35, 99, 224, 440, 783, 1295, 2024, 3024, 4355, 6083, 8280, 11024, 14399, 18495, 23408, 29240, 36099, 44099, 53360, 64008, 76175, 89999, 105624, 123200, 142883, 164835, 189224, 216224, 246015, 278783, 314720, 354024, 396899, 443555, 494208

Offset: 1

Author

Vincenzo Librandi, Nov 30 2009

Keywords

• nonn
• easy

Comments

The products of two consecutive numbers in this sequence may be evaluated in terms of the Frobenius numbers for 5 consecutive integers, A138985(n) = F(n): for n>0, a(2n-1)*a(2n) = F(4n^2-2)^2 - (2n)^2; a(2n)*a(2n+1) = F(4n^2+4n)^2 - (2n+1)^2. - Charlie Marion, Jan 23 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000578 (first differences), A000217, A000537, A138985.

Programs

Magma

[(n-1)*(n+2)*(n^2+n+2)/4: n in [1..50]];

Mathematica

s=0;lst={s};Do[s+=n^3;AppendTo[lst,s],{n,2,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 27 2010 *)
Table[(n-1)*(n+2)*(n^2 + n + 2)/4, {n,1,25}] (* G. C. Greubel, Jul 26 2016 *)
LinearRecurrence[{5,-10,10,-5,1},{0,8,35,99,224},40] (* Harvey P. Dale, Jan 21 2019 *)

PARI

a(n)=(n-1)*(n+2)*(n^2+n+2)/4 \\ Charles R Greathouse IV, Jul 26 2016

Formula

G.f.: x^2*(-8 + 5*x - 4*x^2 + x^3)/(x-1)^5. - R. J. Mathar, Jan 04 2011

a(n) = A000217(n)^2 - 1 = A000537(n)-1. - Charlie Marion, Sep 27 2011

From G. C. Greubel, Jul 26 2016: (Start)

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

E.g.f.: (1/4)*(4 - (4 - 4*x + 14*x^2 + 8*x^3)*exp(x)). (End)

Sum_{n>=2} 1/a(n) = 49/36 - tanh(sqrt(7)*Pi/2)*Pi/sqrt(7). - Amiram Eldar, Mar 02 2023