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%I M4141 N1718 #213 Dec 23 2023 08:25:24
%S 0,1,6,21,55,120,231,406,666,1035,1540,2211,3081,4186,5565,7260,9316,
%T 11781,14706,18145,22155,26796,32131,38226,45150,52975,61776,71631,
%U 82621,94830,108345,123256,139656,157641,177310,198765,222111,247456,274911,304590
%N Doubly triangular numbers: a(n) = n*(n+1)*(n^2+n+2)/8.
%C 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).
%C 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. - [Gupta, 1968, page 653; Bell, 1970, page 279]. - Peter Bertok (peter(AT)bertok.com), Jan 12 2002. See A005045 for another version.
%C 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
%C 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 Janjic_, Jul 30 2007
%C Starting with offset 1 = binomial transform of [1, 5, 10, 9, 3, 0, 0, 0, ...]. - _Gary W. Adamson_, Aug 05 2009
%C Starting with "1" = row sums of triangle A178238. - _Gary W. Adamson_, May 23 2010
%C 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. - _William A. Tedeschi_, Aug 18 2010
%C Partial sums of A006003. - _Jeremy Gardiner_, Jun 23 2013
%C Doubly triangular numbers are revealed in the sums of row sums of Floyd's triangle.
%C 1, 1+5, 1+5+15, ...
%C 1
%C 2 3
%C 4 5 6
%C 7 8 9 10
%C 11 12 13 14 15
%C - _Tony Foster III_, Nov 14 2015
%C From _Jaroslav Krizek_, Mar 04 2017: (Start)
%C For n>=1; a(n) = sum of the different sums of elements of all the nonempty subsets of the sets of numbers from 1 to n.
%C Example: for n = 6; nonempty subsets of the set of numbers from 1 to 3: {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}; sums of elements of these subsets: 1, 2, 3, 3, 4, 5, 6; different sums of elements of these subsets: 1, 2, 3, 4, 5, 6; a(3) = (1+2+3+4+5+6) = 21, ... (End)
%C a(n) is also the number of 4-cycles in the (n+4)-path complement graph. - _Eric W. Weisstein_, Apr 11 2018
%D A. Björner, 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.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 124, #25, Q(3,r).
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D R. P. Stanley, Enumerative Combinatorics I, p. 292.
%H T. D. Noe and William A. Tedeschi, <a href="/A002817/b002817.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms computed by T. D. Noe)
%H G. E. Andrews, P. Paule, A. Riese and V. Strehl, <a href="http://www.risc.uni-linz.ac.at/research/combinat/risc/publications/#ppaule">MacMahon's partition analysis V. Bijections, recursions and magic squares</a>, p. 37.
%H Weymar Astaiza, Alexander J. Barrios, Henry Chimal-Dzul, Stephan Ramon Garcia, Jaaziel de la Luz, Victor H. Moll, Yunied Puig, and Diego Villamizar, <a href="https://arxiv.org/abs/2309.13741">Symmetric tensor powers of graphs</a>, arXiv:2309.13741 [math.CO], 2023. See p. 12.
%H Matthias Beck, <a href="https://arxiv.org/abs/math/0201013">The number of "magic" squares and hypercubes</a>, arXiv:math/0201013 [math.CO], 2002-2005.
%H A. G. Bell, <a href="http://dx.doi.org/10.1093/comjnl/13.3.278">Partitioning integers in n dimensions</a>, The Computer Journal, 13 (1970), 278-283.
%H Miklos Bona, <a href="http://www.jstor.org/stable/2691261">A New Proof of the Formula for the Number of 3 X 3 Magic Squares</a>, Mathematics Magazine, Vol. 70, No. 3 (Jun., 1997), pp. 201-203.
%H L. Carlitz, <a href="https://projecteuclid.org/euclid.dmj/1077376694#ui-tabs-1">Enumeration of symmetric arrays</a>, Duke Math. J., Vol. 33(4) (1966), pp. 771-782.
%H Brian Conrey and Alex Gamburd, <a href="http://dx.doi.org/10.1016/j.jnt.2005.01.006">Pseudomoments of the Riemann zeta-function and pseudomagic squares</a>, Journal of Number Theory, Volume 117, Issue 2, April 2006, Pages 263-278. See H4 on p. 269.
%H P. Diaconis and A. Gamburd, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i2r2">Random matrices, magic squares and matching polynomials</a>, Volume 11, Issue 2 (2004-6) (The Stanley Festschrift volume), Research Paper #R2.
%H Robert W. Donley, <a href="https://arxiv.org/abs/1911.00977">Partitions for semi-magic squares of size three</a>, arXiv:1911.00977 [math.CO], 2019.
%H I. J. Good, <a href="http://www.jstor.org/stable/2958586">On the application of symmetric Dirichlet distributions and their mixtures to contingency tables</a>, Ann. Statist. 4 (1976), no. 6, 1159-1189.
%H I. J. Good, <a href="/A001496/a001496_1.pdf">On the application of symmetric Dirichlet distributions and contingency tables</a>, pp. 1178-1179. (Annotated scanned copy)
%H Hansraj Gupta, <a href="https://doi.org/10.1215/S0012-7094-68-03567-9">Enumeration of symmetric matrices</a>, Duke Math. J. 35 (3), 653-659, (September 1968).
%H D. M. Jackson and G. H. J. van Rees, <a href="http://dx.doi.org/10.1137/0204040">The enumeration of generalized double stochastic nonnegative integer square matrices</a>, SIAM J. Comput., 4 (1975), 474-477.
%H D. M. Jackson & G. H. J. van Rees, <a href="/A002817/a002817.pdf">The enumeration of generalized double stochastic nonnegative integer square matrices</a>, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Neven Juric, <a href="/A002817/a002817.ps">Illustration of the 55 3 X 3 matrices</a>
%H Michal Opler, Pavel Valtr, and Tung Anh Vu, <a href="https://dccg.upc.edu/eurocg23/wp-content/uploads/2023/03/Session-7B-Talk-1.pdf">On the Arrangement of Hyperplanes Determined by n Points</a>, EuroCG (39th European Workshop on Computational Geometry, Barcelona, Spain 2023) Session 7B, Talk 1, Vol. 54, No. 6.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Henry Warburton, <a href="https://archive.org/stream/transactionsofca09camb#page/470/mode/2up">On Self-Repeating Series</a>, Transactions of the Cambridge Philosophical Society, Vol. 9, 471-486, 1856.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PathComplementGraph.html">Path Complement Graph</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = 3*binomial(n+2, 4) + binomial(n+1, 2).
%F G.f.: x*(1 + x + x^2)/(1-x)^5. - _Simon Plouffe_ (in his 1992 dissertation); edited by _N. J. A. Sloane_, May 13 2008
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 3. - _Warut Roonguthai_, Dec 13 1999
%F 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
%F a(n) = Sum(Sum(1 + Sum(3*n))). - Xavier Acloque, Jan 21 2003
%F a(n) = A000332(n+1) + A000332(n+2) + A000332(n+3), with A000332(n) = binomial(n, 4). - _Mitch Harris_, Oct 17 2006 and _Bruce J. Nicholson_, Oct 22 2017
%F a(n) = Sum_{i=1..C(n,2)} i = C(C(n,2) + 1, 2) = A000217(A000217(n+1)). - _Enrique Pérez Herrero_, Jun 11 2012
%F Euler transform of length 3 sequence [6, 0, -1]. - _Michael Somos_, Nov 19 2015
%F E.g.f.: x*(8 + 16*x + 8*x^2 + x^3)*exp(x)/8. - _Ilya Gutkovskiy_, Apr 26 2016
%F Sum_{n>=1} 1/a(n) = 6 - 4*Pi*tanh(sqrt(7)*Pi/2)/sqrt(7) = 1.25269064911978447... . - _Vaclav Kotesovec_, Apr 27 2016
%F a(n) = A000217(n)*A000124(n)/2.
%F a(n) = ((n-1)^4 + 3*(n-1)^3 + 2*(n-1)^2 + 2*n))/8. - _Bruce J. Nicholson_, Apr 05 2017
%F a(n) = (A016754(n)+ A007204(n)- 2) / 32. - _Bruce J. Nicholson_, Apr 14 2017
%F a(n) = a(-1-n) for all n in Z. - _Michael Somos_, Apr 17 2017
%F a(n) = T(T(n)) where T are the triangular numbers A000217. - _Albert Renshaw_, Jan 05 2020
%F a(n) = 2*n^2 - n + 6*binomial(n, 3) + 3*binomial(n, 4). - _Ryan Jean_, Mar 20 2021
%e G.f. = x + 6*x^2 + 21*x^3 + 55*x^4 + 120*x^5 + 231*x^6 + 406*x^7 + 666*x^8 + ...
%p A002817 := n->n*(n+1)*(n^2+n+2)/8;
%t a[ n_] := n (n + 1) (n^2 + n + 2) / 8; (* _Michael Somos_, Jul 24 2002 *)
%t LinearRecurrence[{5,-10,10,-5,1}, {0,1,6,21,55},40] (* _Harvey P. Dale_, Jul 18 2011 *)
%t nn=50;Join[{0},With[{c=(n(n+1))/2},Flatten[Table[Take[Accumulate[Range[ (nn(nn+1))/2]], {c,c}],{n,nn}]]]] (* _Harvey P. Dale_, Mar 19 2013 *)
%o (PARI) {a(n) = n * (n+1) * (n^2 + n + 2) / 8}; /* _Michael Somos_, Jul 24 2002 */
%o (PARI) concat(0, Vec(x*(1+x+x^2)/(1-x)^5 + O(x^50))) \\ _Altug Alkan_, Nov 15 2015
%Y Cf. A006003 (first differences), A165211 (mod 2).
%Y Multiple triangular: A000217, A064322, A066370.
%Y Cf. A001496, A001621, A178238, A005045, A002721.
%Y Cf. A006528 (square colorings).
%Y Cf. A236770 (see crossrefs).
%Y Cf. A016754, A007204.
%Y Row n=3 of A257493 and row n=2 of A331436 and A343097.
%Y Cf. A000332.
%Y Cf. A000292 (3-cycle count of \bar P_{n+4}), A060446 (5-cycle count of \bar P_{n+3}), A302695 (6-cycle count of \bar P_{n+5}).
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999
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