A100119 a(n) = n-th centered n-gonal number.

1, 2, 7, 19, 41, 76, 127, 197, 289, 406, 551, 727, 937, 1184, 1471, 1801, 2177, 2602, 3079, 3611, 4201, 4852, 5567, 6349, 7201, 8126, 9127, 10207, 11369, 12616, 13951, 15377, 16897, 18514, 20231, 22051, 23977, 26012, 28159, 30421, 32801, 35302

Offset: 0

Author

Jonathan Vos Post, Dec 26 2004

Keywords

• easy
• nonn

Comments

a(n) is n times the n-th triangular number plus 1. - Thomas M. Green, Nov 16 2009

From Gary W. Adamson, Jul 31 2010: (Start)

Equals (1, 2, 3, 4, ...) convolved with (1, 0, 4, 7, 10, 13, ...).

Example: a(5) = 76 = (6, 5, 4, 3, 2, 1) dot (1, 0, 4, 7, 10, 13) = (6 + 0 + 16 + 21 + 20 + 13). (End)

Examples

			a(2) = 2*3 + 1 = 7, a(3) = 3*6 + 1 = 19, a(4) = 4*10 + 1 = 41. - _Thomas M. Green_, Nov 16 2009
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

See also A101357 (Cumulative sums of the n-th n-gonal numbers).

A diagonal of A101321.

Cf. A060354, A098547, A101357.

Cf. A006003, A005920.

Programs

Magma

I:=[1, 2, 7, 19]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 25 2012

Mathematica

Table[(n^3+n^2)/2+1,{n,0,6!}] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2010 *)
LinearRecurrence[{4,-6,4,-1},{1,2,7,19},40] (* Vincenzo Librandi, Jun 25 2012 *)

PARI

a(n) = n^2*(n+1)/2+1; \\ Altug Alkan, Sep 21 2018

Formula

a(n) = 1 + n*(n + n^2)/2 = 1 + (1/2)*n^2 + (1/2) * n^3 = 1 + mean(n^2, n^3). - Joshua Zucker, May 03 2006

Equals A002411(n) + 1. - Olivier Gérard, Jun 20 2007

G.f.: (1 - 2*x + 5*x^2 - x^3) / (x-1)^4. - R. J. Mathar, Apr 04 2012

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 25 2012

a(n) = (A098547(n)+1)/2. - Richard Turk, Jul 18 2017

a(n) = A060354(n+2) - A000290(n+1) = A006003(n+1) - A005563(n) and for n>0 A005920(n) - A068601(n+1). - Bruce J. Nicholson, Jun 23 2018

Extensions

Corrected and extended by Joshua Zucker, May 03 2006