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

Author

N. J. A. Sloane, May 03 2000

Keywords

• nonn
• easy

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).

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.

Programs

Mathematica

LinearRecurrence[{3, -3, 1}, {1, 7, 22}, 50] (* Paolo Xausa, Jul 03 2025 *)

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

Sage

a = lambda n: hypergeometric([-n, -2], [1], 3)
print([simplify(a(n)) for n in range(46)]) # Peter Luschny, Nov 19 2014

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

Extensions

More terms from James Sellers, May 03 2000

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