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

0,2

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)

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).

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

Example

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

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 *)

Prog

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

Crossrefs

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

A diagonal of A101321.

Cf. A060354, A098547, A101357.

Cf. A006003, A005920.

Sequence in context: A140610 A152461 A215208 * A322385 A350170 A220697

Adjacent sequences: A100116 A100117 A100118 * A100120 A100121 A100122

Keyword

easy,nonn

Author

Jonathan Vos Post, Dec 26 2004

Extensions

Corrected and extended by Joshua Zucker, May 03 2006

Status

approved