A038764 a(n) = (9*n^2 + 3*n + 2)/2.

1, 7, 22, 46, 79, 121, 172, 232, 301, 379, 466, 562, 667, 781, 904, 1036, 1177, 1327, 1486, 1654, 1831, 2017, 2212, 2416, 2629, 2851, 3082, 3322, 3571, 3829, 4096, 4372, 4657, 4951, 5254, 5566, 5887, 6217, 6556, 6904, 7261, 7627, 8002, 8386, 8779, 9181

Offset

0,2

Comments

Coefficients of x^2 of certain rook polynomials (for n>=1; see p. 18 of the Riordan paper). - Emeric Deutsch, Mar 08 2004

a(n) is also the least weight of self-conjugate partitions having n+1 different parts such that each part is congruent to 1 modulo 3. The first such self-conjugate partitions, corresponding to a(n) = 0, 1, 2, 3, are 1, 4+3, 7+4+4+4+3, 10+7+7+7+4+4+4+3. - Augustine O. Munagi, Dec 18 2008

References

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.

A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = binomial(n,0) + 6*binomial(n,1) + 9*binomial(n,2).

From Paul Barry, Mar 15 2003: (Start)

G.f.: (1 + 2*x)^2/(1 - x)^3.

Binomial transform of (1, 6, 9, 0, 0, 0, ...). (End)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - Colin Barker, Jan 22 2018

a(n) = a(n-1) + 3*(3*n-1) for n>0, a(0)=1. - Vincenzo Librandi, Nov 17 2010

a(n) = hypergeometric([-n, -2], [1], 3). - Peter Luschny, Nov 19 2014

E.g.f.: exp(x)*(2 + 12*x + 9*x^2)/2. - Stefano Spezia, Mar 07 2023

Prog

(Sage)
a = lambda n: hypergeometric([-n, -2], [1], 3)
print([simplify(a(n)) for n in range(46)]) # Peter Luschny, Nov 19 2014
(PARI) a(n)=n*(9*n+3)/2+1 \\ Charles R Greathouse IV, Jun 17 2017
(PARI) Vec((1 + 2*x)^2 / (1 - x)^3 + O(x^60)) \\ Colin Barker, Jan 22 2018

Crossrefs

Reflection of A060544 in A081272.

Second column of A024462. Also = A064641(n+1, 2).

Shallow diagonal of triangular spiral in A051682.

Cf. A027468, A080855, A283394.

Partial sums of A122709.

Sequence in context: A033954 A159227 A081274 * A132438 A010001 A197059

Adjacent sequences: A038761 A038762 A038763 * A038765 A038766 A038767

Keyword

nonn,easy

Author

N. J. A. Sloane, May 03 2000

Extensions

More terms from James A. Sellers, May 03 2000

Entry revised by N. J. A. Sloane, Jan 23 2018

Status

approved