A080855 a(n) = (9*n^2 - 3*n + 2)/2.

1, 4, 16, 37, 67, 106, 154, 211, 277, 352, 436, 529, 631, 742, 862, 991, 1129, 1276, 1432, 1597, 1771, 1954, 2146, 2347, 2557, 2776, 3004, 3241, 3487, 3742, 4006, 4279, 4561, 4852, 5152, 5461, 5779, 6106, 6442, 6787, 7141, 7504, 7876, 8257, 8647, 9046

Offset: 0

Author

Paul Barry, Feb 23 2003

Keywords

• nonn
• easy

Comments

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

Row T(3,n) of A080853.

Equals binomial transform of [1, 3, 9, 0, 0, 0, ...] - Gary W. Adamson, Apr 30 2008

a(n) is also the least weight of self-conjugate partitions having n different parts such that each part is congruent to 2 modulo 3. The first such self-conjugate partitions, corresponding to a(n)=1,2,3,4, are 2+2, 5+5+2+2+2, 8+8+5+5+5+2+2+2, 11+11+8+8+8+5+5+5+2+2+2. - Augustine O. Munagi, Dec 18 2008

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=3, (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

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
• A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501.
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
• Leo Tavares, Illustration: Hexagonal Tri-Rays
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A027468, A038764.

Cf. A283394 (see Crossrefs section).

Cf. A003215, A000217, A227776.

Programs

GAP

List([0..50],n->(9*n^2-3*n+2)/2); # Muniru A Asiru, Nov 02 2018

Magma

[(9*n^2 - 3*n +2)/2: n in [0..50]]; // G. C. Greubel, Nov 02 2018

Maple

seq((9*n^2-3*n+2)/2,n=0..50); # Muniru A Asiru, Nov 02 2018

Mathematica

s = 1; lst = {s}; Do[s += n + 2; AppendTo[lst, s], {n, 1, 500, 9}]; lst (* Zerinvary Lajos, Jul 11 2009 *)
Table[(9n^2-3n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1}, {1,4,16}, 50] (* Harvey P. Dale, Jul 24 2013 *)

PARI

a(n)=binomial(3*n,2)+1 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: (1 + x + 7*x^2)/(1 - x)^3.

a(n) = 9*n + a(n-1) - 6 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010

a(n) = n*A005448(n+1) - (n-1)*A005448(n), with A005448(0)=1. - Bruno Berselli, Jan 15 2013

a(0)=1, a(1)=4, a(2)=16; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jul 24 2013

a(n) = A152947(3*n+1). - Franck Maminirina Ramaharo, Jan 10 2018

E.g.f.: (2 + 6*x + 9*x^2)*exp(x)/2. - G. C. Greubel, Nov 02 2018

From Leo Tavares, Feb 20 2022: (Start)

a(n) = A003215(n-1) + 3*A000217(n). See Hexagonal Tri-Rays illustration in links.

a(n) = A227776(n) - 3*A000217(n). (End)

Extensions

Definition replaced with the closed form by Bruno Berselli, Jan 15 2013