A080956 a(n) = (n+1)*(2-n)/2.

1, 1, 0, -2, -5, -9, -14, -20, -27, -35, -44, -54, -65, -77, -90, -104, -119, -135, -152, -170, -189, -209, -230, -252, -275, -299, -324, -350, -377, -405, -434, -464, -495, -527, -560, -594, -629, -665, -702, -740, -779, -819, -860, -902, -945, -989, -1034, -1080, -1127, -1175, -1224, -1274, -1325, -1377

Offset: 0

Author

Paul Barry, Mar 01 2003

Keywords

• sign
• easy

Comments

Coefficient of x in the polynomial C(n,0)+C(n+1,1)x+C(n+2,2)x(x-1)/2.

Equals A154990 * [1,2,3,...]. - Gary W. Adamson & Mats Granvik, Jan 19 2009

a(n) is essentially the case 1 of the polygonal numbers. The polygonal numbers are defined as P_k(n) = Sum_{i=1..n} ((k-2)*i-(k-3)). Thus P_1(n) = n*(3-n)/2 and a(n) = P_1(n+1). See A005563 for the case k=0. - Peter Luschny, Jul 08 2011

This is the case k=-1 of the formula (k*m*(m+1)-(-1)^k+1)/2. See similar sequences listed in A262221. - Bruno Berselli, Sep 17 2015

Examples

			a(5) = 6-(1+2+3+4+5). - _Stanislav Sykora_, Feb 19 2014
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000027, A000096, A000217, A154990, A214292, A262221.

Programs

Magma

[(n+1)*(2-n)/2: n in [0..80]]; // Vincenzo Librandi, Jul 08 2011

Maple

G(x):=exp(x)*(x-x^2/2): f[0]:=G(x): for n from 1 to 54 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=1..54 ); # Zerinvary Lajos, Apr 05 2009

Mathematica

FoldList[#1 - #2 &, 1, Range[0, 44]] (* Arkadiusz Wesolowski, May 26 2013 *)
LinearRecurrence[{3,-3,1},{1,1,0},60] (* Harvey P. Dale, Nov 29 2019 *)

PARI

a(n)=(n+1)*(2-n)/2;

SageMath

def A080956(n): return (2-n)*(n+1)//2 # G. C. Greubel, May 08 2025

Formula

a(n) = 2*(C(n+1, 1)-C(n+2, 2)) = (n+1)*(2-n)/2.

G.f.: (1-2*x)/(1-x)^3. - R. J. Mathar, Jun 11 2009

If we define f(n,i,a) = Sum_{k=0..n-i} (binomial(n,k)*stirling1(n-k,i)*Product_{j=0..k-1} (-a-j)), then a(n) = f(n,n-1,2), for n>=3. - Milan Janjic, Dec 20 2008

E.g.f.: exp(x)*(1-x^2/2). - Zerinvary Lajos, Apr 05 2009, R. J. Mathar, Jun 11 2009

a(n) = - A214292(n,1) for n > 0. - Reinhard Zumkeller, Jul 12 2012

Recurrence: a(0)=1, a(n+1) = a(n) - n. Also a(n)=(n+1)-Sum[k=1..n](k). Also a(n) = A000027(n+1) - A000217(n). Also, for n>1, a(n) = - A000096(n-2). - Stanislav Sykora, Feb 19 2014

Sum_{n>=3} 1/a(n) = -11/9. - Amiram Eldar, Sep 26 2022

Extensions

Lajos e.g.f. adapted to offset zero by R. J. Mathar, Jun 11 2009