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%I #50 Mar 07 2023 18:48:37
%S 1,7,22,46,79,121,172,232,301,379,466,562,667,781,904,1036,1177,1327,
%T 1486,1654,1831,2017,2212,2416,2629,2851,3082,3322,3571,3829,4096,
%U 4372,4657,4951,5254,5566,5887,6217,6556,6904,7261,7627,8002,8386,8779,9181
%N a(n) = (9*n^2 + 3*n + 2)/2.
%C Coefficients of x^2 of certain rook polynomials (for n>=1; see p. 18 of the Riordan paper). - _Emeric Deutsch_, Mar 08 2004
%C 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
%D J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
%H Colin Barker, <a href="/A038764/b038764.txt">Table of n, a(n) for n = 0..1000</a>
%H S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, <a href="https://hrcak.srce.hr/177109">Unbranched catacondensed polygonal systems containing hexagons and tetragons</a>, Croatica Chem. Acta, 69 (1996), 757-774.
%H A. O. Munagi, <a href="http://dx.doi.org/10.1016/j.disc.2007.05.022">Pairing conjugate partitions by residue classes</a>, Discrete Math., 308 (2008), 2492-2501.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = binomial(n,0) + 6*binomial(n,1) + 9*binomial(n,2).
%F From _Paul Barry_, Mar 15 2003: (Start)
%F G.f.: (1 + 2*x)^2/(1 - x)^3.
%F Binomial transform of (1, 6, 9, 0, 0, 0, ...). (End)
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n>2. - _Colin Barker_, Jan 22 2018
%F a(n) = a(n-1) + 3*(3*n-1) for n>0, a(0)=1. - _Vincenzo Librandi_, Nov 17 2010
%F a(n) = hypergeometric([-n, -2], [1], 3). - _Peter Luschny_, Nov 19 2014
%F E.g.f.: exp(x)*(2 + 12*x + 9*x^2)/2. - _Stefano Spezia_, Mar 07 2023
%o (Sage)
%o a = lambda n: hypergeometric([-n, -2], [1], 3)
%o print([simplify(a(n)) for n in range(46)]) # _Peter Luschny_, Nov 19 2014
%o (PARI) a(n)=n*(9*n+3)/2+1 \\ _Charles R Greathouse IV_, Jun 17 2017
%o (PARI) Vec((1 + 2*x)^2 / (1 - x)^3 + O(x^60)) \\ _Colin Barker_, Jan 22 2018
%Y Reflection of A060544 in A081272.
%Y Second column of A024462. Also = A064641(n+1, 2).
%Y Shallow diagonal of triangular spiral in A051682.
%Y Cf. A027468, A080855, A283394.
%Y Partial sums of A122709.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, May 03 2000
%E More terms from _James A. Sellers_, May 03 2000
%E Entry revised by _N. J. A. Sloane_, Jan 23 2018
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