A143689 a(n) = (3*n^2 - n + 2)/2.

1, 2, 6, 13, 23, 36, 52, 71, 93, 118, 146, 177, 211, 248, 288, 331, 377, 426, 478, 533, 591, 652, 716, 783, 853, 926, 1002, 1081, 1163, 1248, 1336, 1427, 1521, 1618, 1718, 1821, 1927, 2036, 2148, 2263, 2381, 2502, 2626, 2753, 2883, 3016, 3152, 3291

Offset: 0

Author

Gary W. Adamson, Aug 29 2008

Keywords

• nonn
• easy

Comments

Equals left border of triangle A033292.

Equals binomial transform of [1, 1, 3, 0, 0, 0, ...].

A242357(a(n)) = 1. - Reinhard Zumkeller, May 11 2014

These might be called "trisected pentagonal numbers": A figurate pentagonal number is composed of three triangles, of which the central one is the largest, and the removal of the triangular frame (3*n) of the central triangle trisects the figure. This is reflected in the formula a(n) = A000326(n+1) - 3*n. See illustration in links. - John Elias, May 27 2022

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036 [math.CO], 2015-2016.
• John Elias, Trisected Pentagonal Numbers
• Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

a(n) = A000326(n+1) - 3n. Third column of A107111.

Cf. A033292, A104249.

Programs

Haskell

a143689 n = n*(3*n-1) `div` 2 + 1 -- Reinhard Zumkeller, May 11 2014

Mathematica

Table[(3n^2-n+2)/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,2,6},50] (* Harvey P. Dale, May 05 2014 *)

Maxima

makelist(binomial(n, 2) + n^2 + 1, n, 0, 100); /* Franck Maminirina Ramaharo, Mar 01 2018 */

PARI

a(n)=(3*n^2-n+2)/2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = A000326(n+1) - 3*n. (A000326 are the pentagonal numbers.)

a(n) = (3*n^2 - n + 2)/2 = A027599(n+1)/2. - R. J. Mathar, Sep 03 2008

a(n) = a(n-1) + 3*n - 2 (with a(0)=1). - Vincenzo Librandi, Nov 25 2010

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

O.g.f.: (1-x+3*x^2)/((1-x)^3). - Eric Werley, Jun 27 2011

a(n) = A104249(-n). - Bruno Berselli, Jul 08 2015

a(n) = binomial(n,2) + n^2 + 1 = A152947(n+1) + A000290(n). - Franck Maminirina Ramaharo, Mar 01 2018

E.g.f.: exp(x)*(2 + 2*x + 3*x^2)/2. - Stefano Spezia, Apr 19 2025

Extensions

Index of A000326 in definition, formula and example corrected by R. J. Mathar, Sep 03 2008