A153977 One-fourth of partial sums of A153976.

2, 9, 27, 65, 135, 252, 434, 702, 1080, 1595, 2277, 3159, 4277, 5670, 7380, 9452, 11934, 14877, 18335, 22365, 27027, 32384, 38502, 45450, 53300, 62127, 72009, 83027, 95265, 108810, 123752, 140184, 158202, 177905, 199395, 222777, 248159, 275652, 305370, 337430

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Keywords

• nonn
• easy
• changed

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000217, A000537, A000578, A003215, A005898, A006007, A027602, A153976.

Programs

Maple

A153977:=n->(1/4)*sum(i^3 + (i+2)^3, i=0..n): seq(A153977(n), n=0..50); # Wesley Ivan Hurt, Feb 04 2017

Mathematica

a[n_]:=n^3;lst={};s=0;Do[s+=(a[n]+a[n+2]);AppendTo[lst,s/4],{n,0,6!}];lst
Accumulate[Array[#^3+(#+2)^3&,40,0]]/4 (* or *) LinearRecurrence[ {5,-10,10,-5,1},{2,9,27,65,135},40] (* Harvey P. Dale, Aug 02 2011 *)

PARI

a(n)=(n^4 + 2*n^3 + 7*n^2 + 6*n)/8 \\ Charles R Greathouse IV, Feb 06 2017

Formula

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5, a(1)=2, a(2)=9, a(3)=27, a(4)=65, a(5)=135. - Harvey P. Dale, Aug 02 2011

a(n) = (A000217(n-1)^2 + A000217(n+1)^2 - 1)/4. - Richard R. Forberg, Dec 25 2013

Recurrence: (n-1)*(n^2 - n + 6)*a(n) = (n+1)*(n^2 + n + 6)*a(n-1). - Vaclav Kotesovec, Dec 26 2013

a(n) = A000217(A000217(n)) + A000217(n). - Bruno Berselli, May 28 2015

a(n) = (A000217(n)^2 + 3*A000217(n))/2 where A000217(n) is the n-th triangular number. - Frederic Isenmann, Feb 04 2017

Sum_{n>=1} 1/a(n) = 14/9 - 4*tanh(sqrt(23)*Pi/2)*Pi/(3*sqrt(23)). - Amiram Eldar, Aug 23 2022

From Elmo R. Oliveira, Aug 28 2025: (Start)

G.f.: x*(2 - x + 2*x^2)/(1-x)^5.

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