A080857 a(n) = (25*n^2 - 15*n + 2)/2.

1, 6, 36, 91, 171, 276, 406, 561, 741, 946, 1176, 1431, 1711, 2016, 2346, 2701, 3081, 3486, 3916, 4371, 4851, 5356, 5886, 6441, 7021, 7626, 8256, 8911, 9591, 10296, 11026, 11781, 12561, 13366, 14196, 15051, 15931, 16836, 17766, 18721, 19701

Offset: 0

Author

Paul Barry, Feb 23 2003

Keywords

• nonn
• easy

Comments

The old definition of this sequence was "Generalized polygonal numbers".

Row T(5,n) of A080853.

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]=5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 3, a(n-1)=coeff(charpoly(A,x),x^(n-2)). - Milan Janjic, Jan 27 2010

Links

• Milan Janjić, Hessenberg Matrices and Integer Sequences, Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.8.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Programs

Mathematica

Table[(25n^2-15n+2)/2,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{1,6,36},50] (* Harvey P. Dale, Aug 14 2018 *)

PARI

a(n)=(25*n^2-15*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

G.f.: (1+3*x+21*x^2)/(1-x)^3

a(n) = 25*n + a(n-1) - 20 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010

From Elmo R. Oliveira, Oct 25 2024: (Start)

E.g.f.: exp(x)*(1 + 5*x + 25*x^2/2).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Extensions

Definition replaced with the closed form by Bruno Berselli, Jan 16 2013