OFFSET
0,2
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).
FORMULA
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
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 *)
PROG
(Haskell)
a143689 n = n*(3*n-1) `div` 2 + 1 -- Reinhard Zumkeller, May 11 2014
(PARI) a(n)=(3*n^2-n+2)/2 \\ Charles R Greathouse IV, Oct 07 2015
(Maxima) makelist(binomial(n, 2) + n^2 + 1, n, 0, 100); /* Franck Maminirina Ramaharo, Mar 01 2018 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Aug 29 2008
EXTENSIONS
Index of A000326 in definition, formula and example corrected by R. J. Mathar, Sep 03 2008
STATUS
approved