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A002817
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Doubly triangular numbers: n*(n+1)*(n^2+n+2)/8.
(Formerly M4141 N1718)
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24
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0, 1, 6, 21, 55, 120, 231, 406, 666, 1035, 1540, 2211, 3081, 4186, 5565, 7260, 9316, 11781, 14706, 18145, 22155, 26796, 32131, 38226, 45150, 52975, 61776, 71631, 82621, 94830, 108345, 123256, 139656, 157641, 177310, 198765, 222111, 247456
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of inequivalent ways to color vertices of a square using <= n colors, allowing rotations and reflections. Group is dihedral group D_8 of order 8 with cycle index (1/8)*(x1^4+2*x4+3*x2^2+2*x1^2*x2); setting all x_i = n gives the formula a(n) = (1/8)*(n^4+2*n+3*n^2+2*n^3).
Number of semi-magic 3 X 3 squares with a line sum of n-1. That is, 3 X 3 matrices of nonnegative integers such that row sums and column sums are all equal to n-1. - Peter Bertok (peter(AT)bertok.com), Jan 12 2002. See A005045 for another version.
Also the coefficient h_2 of x^{n-3} in the shelling polynomial h(x)=h_0*x^n-1 + h_1*x^n-2 + h_2*x^n-3 + ... + h_n-1 for the independence complex of the cycle matroid of the complete graph K_n on n vertices (n>=2) - Woong Kook (andrewk(AT)math.uri.edu), Nov 01 2006
If X is an n-set and Y a fixed 3-subset of X then a(n-4) is equal to the number of 5-subsets of X intersecting Y. - Milan R. Janjic (agnus(AT)blic.net), Jul 30 2007
Starting with offset 1 = binomial transform of [1, 5, 10, 9, 3, 0, 0, 0,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 05 2009]
Starting with "1" = row sums of triangle A178238 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 23 2010]
The equation n*(n+1)*(n^2+n+2)/8 may be arrived at by solving for x in the following equality: (n^2+n)/2 = (sqrt(8x+1)-1)/2. [From William A. Tedeschi (fynmun(AT)att.net), Aug 18 2010]
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REFERENCES
| A. Bj\"orner, The homology and shellability of matroids and geometric lattices, in Matroid Applications (ed. N. White), Encyclopedia of Mathematics and Its Applications, 40, Cambridge Univ. Press 1992.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 124, #25, Q(3,r).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics I, p. 292.
Warburton, Henry. "On Self-Repeating Series." Transactions of the Cambridge Philosophical Society, Vol. 9, 471-486, 1856.
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LINKS
| William A. Tedeschi, Table of n, a(n) for n=1..10000 [This replaces an earlier b-file having 1000 terms computed by T. D. Noe]
G. E. Andrews, P. Paule, A. Riese and V. Strehl, MacMahon's partition analysis V. Bijections, recursions and magic squares, p. 37.
Matthias Beck, The number of "magic" squares and hypercubes
P. Diaconis and A. Gamburd, Random matrices, magic squares and matching polynomials
I. J. Good, On the application of symmetric Dirichlet distributions and their mixtures to contingency tables, Ann. Statist. 4 (1976), no. 6, 1159-1189.
D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477.
Milan Janjic, Two Enumerative Functions
Neven Juric, Illustration of the 55 3 X 3 matrices
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| G.f.: x(1+x+x^2)/(1-x)^5. a(n) = 3*binomial(n+2, 4)+binomial(n+1, 2).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 3 - Warut Roonguthai (warut822(AT)yahoo.com) Dec 13 1999
a(n) = 5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+a(n-5) = A000217(A000217(n)). [Ant King, Nov 18 2010].
a(n) = Sum [ Sum ( 1 + Sum (3*n) ) ]. - Xavier Acloque, Jan 21 2003
a(n) = (n+3 choose 4) + (n+2 choose 4) + (n+1 choose 4) - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Oct 17 2006
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MAPLE
| A002817 := n->n*(n+1)*(n^2+n+2)/8;
A002817:=-(1+z+z**2)/(z-1)**5; [S. Plouffe in his 1992 dissertation.]
a:=n->add(n+add(binomial(n, 2), j=1..n), j=2..n):seq(a(n)/4, n=1..38); [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 26 2008]
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MATHEMATICA
| a[n_] := n(n+1)(n^2+n+2)/8; a@Range[0, 37]
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 1, 6, 21, 55}, 40] (* From Harvey P. Dale, Jul 18 2011 *)
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PROG
| (PARI) a(n)=n*(n+1)*(n^2+n+2)/8
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CROSSREFS
| Cf. A000217, A064322, A066370, A001496.
A001621 [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Aug 26 2008]
Cf. A178238 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 23 2010]
Cf. A006528 (square colorings).
Sequence in context: A115052 A025203 A162539 * A132366 A015641 A050190
Adjacent sequences: A002814 A002815 A002816 * A002818 A002819 A002820
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999
Plouffe Maple line edited by N. J. A. Sloane (njas(AT)research.att.com), May 13 2008
Mathematica code edited by Ant King, Nov 18 2010.
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