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 A080855 a(n) = (9*n^2 - 3*n + 2)/2. 9
 1, 4, 16, 37, 67, 106, 154, 211, 277, 352, 436, 529, 631, 742, 862, 991, 1129, 1276, 1432, 1597, 1771, 1954, 2146, 2347, 2557, 2776, 3004, 3241, 3487, 3742, 4006, 4279, 4561, 4852, 5152, 5461, 5779, 6106, 6442, 6787, 7141, 7504, 7876, 8257, 8647, 9046 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The old definition of this sequence was "Generalized polygonal numbers". Row T(3,n) of A080853. Equals binomial transform of [1, 3, 9, 0, 0, 0, ...] - Gary W. Adamson, Apr 30 2008 a(n) is also the least weight of self-conjugate partitions having n different parts such that each part is congruent to 2 modulo 3. The first such self-conjugate partitions, corresponding to a(n)=1,2,3,4, are 2+2, 5+5+2+2+2, 8+8+5+5+5+2+2+2, 11+11+8+8+8+5+5+5+2+2+2. - Augustine O. Munagi, Dec 18 2008 Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=3, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 3, a(n-1) = -coeff(charpoly(A,x), x^(n-2)). - Milan Janjic, Jan 27 2010 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 M. Janjic, Hessenberg Matrices and Integer Sequences , J. Int. Seq. 13 (2010) # 10.7.8 A. O. Munagi, Pairing conjugate partitions by residue classes, Discrete Math., 308 (2008), 2492-2501. 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 G.f.: (1 + x + 7*x^2)/(1 - x)^3. a(n) = 9*n + a(n-1) - 6 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010 a(n) = n*A005448(n+1) - (n-1)*A005448(n), with A005448(0)=1. - Bruno Berselli, Jan 15 2013 a(0)=1, a(1)=4, a(2)=16; for n > 2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jul 24 2013 a(n) = A152947(3*n+1). - Franck Maminirina Ramaharo, Jan 10 2018 MATHEMATICA s = 1; lst = {s}; Do[s += n + 2; AppendTo[lst, s], {n, 1, 500, 9}]; lst (* Zerinvary Lajos, Jul 11 2009 *) Table[(9n^2-3n+2)/2, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {1, 4, 16}, 50] (* Harvey P. Dale, Jul 24 2013 *) PROG (PARI) a(n)=binomial(3*n, 2)+1 \\ Charles R Greathouse IV, Oct 07 2015 CROSSREFS Cf. A027468, A038764. Cf. A283394 (see Crossrefs section). Sequence in context: A173545 A080709 A256322 * A203299 A198015 A103770 Adjacent sequences:  A080852 A080853 A080854 * A080856 A080857 A080858 KEYWORD nonn,easy,changed AUTHOR Paul Barry, Feb 23 2003 EXTENSIONS Definition replaced with the closed form by Bruno Berselli, Jan 15 2013 STATUS approved

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Last modified May 25 10:41 EDT 2018. Contains 304560 sequences. (Running on oeis4.)