%I M0998 N0374
%S 0,0,1,2,4,6,9,12,16,20,25,30,36,42,49,56,64,72,81,90,100,110,121,132,
%T 144,156,169,182,196,210,225,240,256,272,289,306,324,342,361,380,400,
%U 420,441,462,484,506,529,552,576,600,625,650,676,702,729,756,784,812
%N Quartersquares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).
%C b(n) = A002620(n+2) = number of multigraphs with loops on 2 nodes with n edges [so g.f. for b(n) is 1/((1x)^2*(1x^2))]. Also number of 2covers of an nset; also number of 2 X n binary matrices with no zero columns up to row and column permutation.  _Vladeta Jovovic_, Jun 08 2000
%C a(n) is also the maximal number of edges that a trianglefree graph of n vertices can have. For n = 2m, the maximum is achieved by the bipartite graph K(m, m); for n = 2m + 1, the maximum is achieved by the bipartite graph K(m, m + 1).  Avi Peretz (njk(AT)netvision.net.il), Mar 18 2001
%C a(n) is the number of arithmetic progressions of 3 terms and any mean which can be extracted from the set of the first n natural numbers (starting from 1).  _Santi Spadaro_, Jul 13 2001
%C This is also the order dimension of the (strong) Bruhat order on the Coxeter group A_{n1} (the symmetric group S_n).  Nathan Reading (reading(AT)math.umn.edu), Mar 07 2002
%C Let M_n denote the n X n matrix m(i,j) = 2 if i = j; m(i, j) = 1 if (i+j) is even; m(i, j) = 0 if i + j is odd, then a(n+2) = det M_n.  _Benoit Cloitre_, Jun 19 2002
%C Sums of pairs of neighboring terms are triangular numbers in increasing order.  _Amarnath Murthy_, Aug 19 2002
%C Also, from the starting position in standard chess, minimum number of captures by pawns of the same color to place n of them on the same file (column). Beyond a(6), the board and number of pieces available for capture are assumed to be extended enough to accomplish this task.  _Rick L. Shepherd_, Sep 17 2002
%C For example, a(2) = 1 and one capture can produce "doubled pawns", a(3) = 2 and two captures is sufficient to produce tripled pawns, etc. (Of course other, uncounted, noncapturing pawn moves are also necessary from the starting position in order to put three or more pawns on a given file.)  _Rick L. Shepherd_, Sep 17 2002
%C Terms are the geometric mean and arithmetic mean of their neighbors alternately.  _Amarnath Murthy_, Oct 17 2002
%C Maximum product of two integers whose sum is n.  _Matthew Vandermast_, Mar 04 2003
%C a(n+1) gives number of nonsymmetric partitions of n into at most 3 parts, with zeros used as padding. E.g., a(6) = 12 because we can write 5 = 5 + 0 + 0 = 0 + 5 + 0 = 4 + 1 + 0 = 1 + 4 + 0 = 1 + 0 + 4 = 3 + 2 + 0 = 2 + 3 + 0 = 2 + 0 + 3 = 2 + 2 + 1 = 2 + 1 + 2 = 3 + 1 + 1 = 1 + 3 + 1.  _Jon Perry_, Jul 08 2003
%C a(n1) gives number of distinct elements greater than 1 of nonsymmetric partitions of n into at most 3 parts, with zeros used as padding, appear in the middle. E.g., 5 = 5 + 0 + 0 = 0 + 5 + 0 = 4 + 1 + 0 = 1 + 4 + 0 = 1 + 0 + 4 = 3 + 2 + 0 = 2 + 3 + 0 = 2 + 0 + 3 = 2 + 2 + 1 = 2 + 1 + 2 = 3 + 1 + 1 = 1 + 3 + 1. Of these, 050, 140, 320, 230, 221, 131 qualify and a(4) = 6.  _Jon Perry_, Jul 08 2003
%C Union of square numbers (A000290) and oblong numbers (A002378).  _Lekraj Beedassy_, Oct 02 2003
%C Conjectured size of the smallest critical set in a Latin square of order n (true for n <= 8).  Richard Bean (rwb(AT)eskimo.com), Jun 12 2003 and Nov 18 2003
%C a(n) gives number of maximal strokes on complete graph K_n, when edges on K_n can be assigned directions in any way. A "stroke" is a locally maximal directed path on a directed graph. Examples: n = 3, two strokes can exist, "x > y > z" and " x > z", so a(3) = 2. n = 4, four maximal strokes exist, "u > x > z" and "u > y" and "u > z" and "x > y > z", so a(4) = 4.  _Yasutoshi Kohmoto_, Dec 20 2003
%C Number of symmetric Dyck paths of semilength n+1 and having three peaks. E.g., a(4) = 4 because we have U*DUUU*DDDU*D, UU*DUU*DDU*DD, UU*DDU*DUU*DD and UUU*DU*DU*DDD, where U = (1, 1), D = (1, 1) and * indicates a peak.  _Emeric Deutsch_, Jan 12 2004
%C Number of valid inequalities of the form j + k < n + 1, where j and k are positive integers, j <= k, n >= 0.  _Rick L. Shepherd_, Feb 27 2004
%C See A092186 for another application.
%C Also, the number of nonisomorphic transversal combinatorial geometries of rank 2.  Alexandr S. Radionov (rasmailru(AT)mail.ru), Jun 02 2004
%C a(n+1) is the transform of n under the Riordan array (1/(1x^2), x).  _Paul Barry_, Apr 16 2005
%C 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ... specifies the largest number of copies of any of the gifts you receive on the nth day in the "Twelve Days of Christmas" song. For example, on the fifth day of Christmas, you have 9 French hens.  _Alonso del Arte_, Jun 17 2005
%C a(n) = Sum_{k=0..n} Min{k, nk}, sums of rows of the triangle in A004197.  _Reinhard Zumkeller_, Jul 27 2005
%C a(n+1) is the number of noncongruent integersided triangles with largest side n.  _David W. Wilson_ [Comment corrected Sep 26 2006]
%C A quartersquare table can be used to multiply integers since n*m = a(n+m)  a(nm) for all integer n, m.  _Michael Somos_, Oct 29 2006
%C The sequence is the size of the smallest strong critical set in a Latin square of order n.  G.H.J. van Rees (vanrees(AT)cs.umanitoba.ca), Feb 16 2007
%C Maximal number of squares (maximal area) in a polyomino with perimeter 2n.  _Tanya Khovanova_, Jul 04 2007
%C For n >= 3 a(n1) is the number of bracelets with n+3 beads, 2 of which are red, 1 of which is blue.  _Washington Bomfim_, Jul 26 2008
%C Equals row sums of triangle A122196.  _Gary W. Adamson_, Nov 29 2008
%C Also a(n) is the number of different patterns of a 2colored 3partition of n.  _Ctibor O. Zizka_, Nov 19 2014
%C Also a(n1) = C(((n+(n mod 2))/2), 2) + C(((n(n mod 2))/2), 2), so this is the second diagonal of A061857 and A061866, and each evenindexed term is the average of its two neighbors.  _Antti Karttunen_
%C Equals triangle A171608 * ( 1, 2, 3, ...).  _Gary W. Adamson_, Dec 12 2009
%C a(n) gives the number of nonisomorphic faithful representations of the Symmetric group S_3 of dimension n. Any faithful representation of S_3 must contain at least one copy of the 2dimensional irrep, along with any combination of the two 1dimensional irreps.  _Andrew Rupinski_, Jan 20 2011
%C a(n+2) gives the number of ways to make change for "c" cents, letting n = floor(c/5) to account for the 5repetitive nature of the task, using only pennies, nickels and dimes (see A187243).  _Adam Sasson_, Mar 07 2011
%C a(n) belongs to the sequence if and only if a(n) = floor(sqrt(a(n))) * ceiling(sqrt(a(n))), that is, a(n) = k^2 or a(n) = k*(k+1), k >= 0.  _Daniel Forgues_, Apr 17 2011
%C a(n) is the sum of the positive integers < n that have the opposite parity as n.
%C Deleting the first 0 from the sequence results in a sequence b = 0, 1, 2, 4, ... such that b(n) is sum of the positive integers <= n that have the same parity as n. The sequence b(n) is the additive counterpart of the double factorial.  _Peter Luschny_, Jul 06 2011
%C Third outer diagonal of Losanitsch's Triangle, A034851.  _Fred Daniel Kline_, Sep 10 2011
%C Written as a(1) = 1, a(n) = a(n1) + ceiling (a(n1)) this is to ceiling as A002984 is to floor, and as A033638 is to round.  _Jonathan Vos Post_, Oct 08 2011
%C a(n2) gives the number of distinct graphs with n vertices and n regions.  _Erik Hasse_, Oct 18 2011
%C Construct the nth row of Pascal's triangle (A007318) from the preceding row, starting with row 0 = 1. a(n) counts the total number of additions required to compute the triangle in this way up to row n, with the restrictions that copying a term does not count as an addition, and that all additions not required by the symmetry of Pascal's triangle are replaced by copying terms.  _Douglas Latimer_, Mar 05 2012
%C a(n) is the sum of the positive differences of the parts in the partitions of n+1 into exactly 2 parts.  _Wesley Ivan Hurt_, Jan 27 2013
%C a(n) is the maximum number of covering relations possible in an nelement graded poset. For n = 2m, this bound is achieved for the poset with two sets of m elements, with each point in the "upper" set covering each point in the "lower" set. For n = 2m+1, this bound is achieved by the poset with m nodes in an upper set covering each of m+1 nodes in a lower set.  _Ben Branman_, Mar 26 2013
%C a(n+2) is the number of (integer) partitions of n into 2 sorts of 1's and 1 sort of 2's.  _Joerg Arndt_, May 17 2013
%C Alternative statement of Oppermann's conjecture: For n>2, there is at least one prime between a(n) and a(n+1).  _Ivan N. Ianakiev_, May 23 2013. [This conjecture was mentioned in A220492, A222030.  _Omar E. Pol_, Oct 25 2013]
%C For any given prime number, p, there are an infinite number of a(n) divisible by p, with those a(n) occurring in evenly spaced clusters of three as a(n), a(n+1), a(n+2) for a given p. The divisibility of all a(n) by p and the result are given by the following equations, where m >= 1 is the cluster number for that p: a(2m*p)/p = p*m^2  m; a(2m*p + 1)/p = p*m^2; a(2m*p + 2)/p = p*m^2 + m. The number of a(n) instances between clusters is 2*p  3.  _Richard R. Forberg_, Jun 09 2013
%C Apart from the initial term this is the elliptic troublemaker sequence R_n(1,2) in the notation of Stange (see Table 1, p.16). For other elliptic troublemaker sequences R_n(a,b) see the cross references below.  _Peter Bala_, Aug 08 2013
%C a(n) is also the total number of twin hearts patterns (6c4c) packing into (n+1) X (n+1) coins, the coins left is A042948 and the voids left is A000982. See illustration in links.  _Kival Ngaokrajang_, Oct 24 2013
%C Partitions of 2n into parts of size 1, 2 or 4 where the largest part is 4, i.e., A073463(n,2).  _Henry Bottomley_, Oct 28 2013
%C a(n+1) is the minimum length of a sequence (of not necessarily distinct terms) that guarantees the existence of a (not necessarily consecutive) subsequence of length n in which like terms appear consecutively. This is also the minimum cardinality of an ordered set S that ensures that, given any partition of S, there will be a subset T of S so that the induced subpartition on T avoids the pattern ac/b, where a < b < c.  _Eric Gottlieb_, Mar 05 2014
%C A237347(a(n)) = 3; A235711(n) = A003415(a(n)).  _Reinhard Zumkeller_, Mar 18 2014
%C Also the number of elements of the list 1..n+1 such that for any two elements {x,y} the integer (x+y)/2 lies in the range ]x,y[.  _Robert G. Wilson v_, May 22 2014
%C Number of lattice points (x,y) inside the region of the coordinate plane bounded by x <= n, 0 < y <= x/2. For a(11)=30 there are exactly 30 lattice points in the region below:
%C 6 .
%C . . 
%C 5 .__+__+
%C . .   
%C 4 .__+__+__+__+
%C . .     
%C 3 .__+__+__+__+__+__+
%C . .       
%C 2 .__+__+__+__+__+__+__+__+
%C . .         
%C 1 .__+__+__+__+__+__+__+__+__+__+
%C ..           
%C 0.__+__+__+__+__+__+__+__+__+__+__+_________
%C 0 1 2 3 4 5 6 7 8 9 10 11 .. n
%C 0 0 1 2 4 6 9 12 16 20 25 30 .. a(n)  _Wesley Ivan Hurt_, Oct 26 2014
%C a(n+1) is the greatest integer k for which there exists an n x n matrix M of nonnegative integers with every row and column summing to k, such that there do not exist n entries of M, all greater than 1, and no two of these entries in the same row or column.  _Richard Stanley_, Nov 19 2014
%C In a tiling of the triangular shape T_N with row length k for row k = 1, 2, ..., N >= 1 (or, alternatively row length N = 1k for row k) with rectangular tiles, there can appear rectangles (i, j), N >= i >= j >= 1, of a(N+1) types (and their transposed shapes obtained by interchanging i and j). See the Feb 27 2004 comment above from _Rick L. Shepherd_. The motivation to look into this came from a proposal of _Kival Ngaokrajang_ in A247139.  _Wolfdieter Lang_, Dec 09 2014
%C Every positive integer is a sum of at most four distinct quartersquares; see A257018.  _Clark Kimberling_, Apr 15 2015
%C a(n+1) gives the maximal number of distinct elements of an n X n matrix which is symmetric (w.r.t. the main diagonal) and symmetric w.r.t. the main antidiagonal. Such matrices are called bisymmetric. See the Wikipedia link.  _Wolfdieter Lang_, Jul 07 2015
%C For 2^a(n+1), n >= 1, the number of binary bisymmetric n X n matrices, see A060656(n+1) and the comment and link by _Dennis P. Walsh_.  _Wolfdieter Lang_, Aug 16 2015
%C a(n) is the number of partitions of 2n+1 of length three with exactly two even entries (see below example).  _John M. Campbell_, Jan 29 2016
%C a(n) is the sum of the asymmetry degrees of all 01avoiding binary words of length n. The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. a(6) = 9 because the 01avoiding binary words of length 6 are 000000, 100000, 110000, 111000, 111100, 111110, and 111111, and the sum of their asymmetry degrees is 0 + 1 + 2 + 3 + 2 + 1 + 0 = 9. Equivalently, a(n) = Sum_{k>=0} k*A275437(n,k).  _Emeric Deutsch_, Aug 15 2016
%C a(n) is the number of ways to represent all the integers in the interval [3,n+1] as the sum of two distinct natural numbers. E.g., a(7)=12 as there are 12 different ways to represent all the numbers in the interval [3,8] as the sum of two distinct parts: 1+2=3, 1+3=4, 1+4=5, 1+5=6, 1+6=7, 1+7=8, 2+3=5, 2+4=6, 2+5=7, 2+6=8, 3+4=7, 3+5=8.  _Anton Zakharov_, Aug 24 2016
%C a(n+2) is the number of conjugacy classes of involutions (considering the identity as an involution) in the hyperoctahedral group C_2 wreath S_n.  _Mark Wildon_, Apr 22 2017
%C a(n+2) is the maximum number of pieces of a pizza that can be made with n cuts that are parallel or perpendicular to each other.  _Anton Zakharov_, May 11 2017
%C Also the matching number of the n X n black bishop graph.  _Eric W. Weisstein_, Jun 26 2017
%C The answer to a question posed by W. Mantel: a(n) is the maximum number of edges in an nvertex trianglefree graph. Also solved by H. Gouwentak, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff.  _Charles R Greathouse IV_, Feb 01 2018
%C Number of nonisomorphic outer planar graphs of order n >= 3, size n+2, and maximum degree 4.  _Christian Barrientos_ and _Sarah Minion_, Feb 27 2018
%D Sergei Abramovich, Combinatorics of the Triangle Inequality: From Straws to Experimental Mathematics for Teachers, Spreadsheets in Education (eJSiE), Vol. 9, Issue 1, Article 1, 2016. See Fig. 3.
%D G. L. Alexanderson et al., The William Powell Putnam Mathematical Competition  Problems and Solutions: 19651984, M.A.A., 1985; see Problem A1 of 27th Competition.
%D T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 1976, page 73, problem 25.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. AddisonWesley, Reading, MA, 1990, p. 99.
%D D. E. Knuth, The art of programming, Vol. 1, 3rd Edition, AddisonWesley, 1997, Ex. 36 of section 1.2.4.
%D J. Nelder, Critical sets in Latin squares, CSIRO Division of Math. and Stats. Newsletter, Vol. 38 (1977), p. 4.
%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).
%H Franklin T. AdamsWatters, <a href="/A002620/b002620.txt">Table of n, a(n) for n = 0..10000</a>
%H Suayb S. Arslan, <a href="https://arxiv.org/abs/1709.07446"> Asymptotically MDS Array BPXOR Codes</a>, arXiv:1709.07949 [cs.IT], 2017.
%H J. A. Bate & G. H. J. van Rees, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.71.2828">The Size of the Smallest Strong Critical Set in a Latin Square</a>, Ars Combinatoria, Vol. 53 (1999) 7383.
%H M. Benoumhani, M. Kolli, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Benoumhani/benoumhani6.html">Finite topologies and partitions</a>, JIS 13 (2010) # 10.3.5, Lemma 6 first line.
%H G. Blom and C.E. Froeberg, <a href="/A002575/a002575.pdf">Om myntvaexling (On moneychanging) [Swedish]</a>, Nordisk Matematisk Tidskrift, 10 (1962), 5569, 103. [Annotated scanned copy] See Table 4, row 3.
%H Washington G. Bomfim, <a href="http://commons.wikimedia.org/wiki/Image:A002620.PNG">Illustration of the bracelets with 8 beads, 2 of which are red, 1 of which is blue.</a>
%H H. Bottomley, <a href="/A002620/a002620.gif">Illustration of initial terms</a>
%H J. Brandts and C. Cihangir, <a href="http://www.math.cas.cz/~am2013/proceedings/contributions/brandts.pdf">Counting triangles that share their vertices with the unit ncube</a>, in Conference Applications of Mathematics 2013 in honor of the 70th birthday of Karel Segeth. Jan Brandts, Sergey Korotov, et al., eds., Institute of Mathematics AS CR, Prague 2013.
%H Jan Brandts, A Cihangir, <a href="http://arxiv.org/abs/1512.03044">Enumeration and investigation of acute 0/1simplices modulo the action of the hyperoctahedral group</a>, arXiv preprint arXiv:1512.03044 [math.CO], 2015.
%H P. J. Cameron, <a href="http://www.maths.qmw.ac.uk/~pjc/bcc/allprobs.pdf">BCC Problem List</a>, Problem BCC15.15 (DM285), Discrete Math. 167/168 (1997), 605615.
%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H Johann Cigler, <a href="https://arxiv.org/abs/1711.03340">Some remarks on RogersSzegö polynomials and Losanitsch's triangle</a>, arXiv:1711.03340 [math.CO], 2017.
%H E. Fix and J. L. Hodges, Jr., <a href="http://www.jstor.org/stable/2236885">Significance probabilities of the Wilcoxon test</a>, Annals Math. Stat., 26 (1955), 301312.
%H E. Fix and J. L. Hodges, <a href="/A000601/a000601.pdf">Significance probabilities of the Wilcoxon test</a>, Annals Math. Stat., 26 (1955), 301312. [Annotated scanned copy]
%H A. Ganesan, <a href="http://arxiv.org/abs/1206.6279">Automorphism groups of graphs</a>, arXiv preprint arXiv:1206.6279 [cs.DM], 2012.  From _N. J. A. Sloane_, Dec 17 2012
%H E. Gottlieb, M. Sheard, <a href="http://discretews.math.msstate.edu/2014/gottlieb_slides.pdf">An ErdosSzekeres result for set partitions</a>, Slides from a talk, Nov 14 2014. [A006260 is a typo for A002620]
%H R. K. Guy, <a href="/A002186/a002186.pdf">Letters to N. J. A. Sloane, JuneAugust 1968</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=105">Encyclopedia of Combinatorial Structures 105</a>
%H O. A. Ivanov, <a href="http://www.jstor.org/stable/10.4169/000298910X523362">On the number of regions into which n straight lines divide the plane</a>, Amer. Math. Monthly, 117 (2010), 881888. See Th. 4.
%H T. Jenkyns and E. Muller, <a href="http://www.jstor.org/stable/2589119">Triangular triples from ceilings to floors</a>, Amer. Math. Monthly, 107 (Aug. 2000), 634639.
%H V. Jovovic, Vladeta Jovovic, <a href="/A005748/a005748.pdf">Number of binary matrices</a>
%H Clark Kimberling and John E. Brown, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Kimberling/kimber67.html">Partial Complements and Transposable Dispersions</a>, J. Integer Seqs., Vol. 7, 2004.
%H S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, <a href="http://arXiv.org/abs/nlin.SI/0104020">Blending two discrete integrability criteria: ...</a>, arXiv:nlin/0104020 [nlin.SI], 2001.
%H W. Lanssens, B. Demoen, P.L. Nguyen, <a href="http://www.cs.kuleuven.ac.be/publicaties/rapporten/cw/CW666.pdf">The Diagonal Latin Tableau and the Redundancy of its Disequalities</a>, Report CW 666, July 2014, Department of Computer Science, KU Leuven.
%H S. M. Losanitsch, <a href="http://dx.doi.org/10.1002/cber.189703002144">Die IsomerieArten bei den Homologen der ParaffinReihe</a>, Chem. Ber. 30 (1897), 19171926.
%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die IsomerieArten bei den Homologen der ParaffinReihe</a>, Chem. Ber. 30 (1897), 19171926. (Annotated scanned copy)
%H W. Mantel and W. A. Wythoff, <a href="https://babel.hathitrust.org/cgi/pt?id=mdp.39015053335595;view=1up;seq=68">Vraagstuk XXVIII</a>, Wiskundige Opgaven, 10 (1907), pp. 6061.
%H Rene Marczinzik, <a href="https://arxiv.org/abs/1701.00972">Finitistic Auslander algebras</a>, arXiv:1701.00972 [math.RT], 2017 [Page 9, Conjecture].
%H Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Merca/merca3.html">Inequalities and Identities Involving Sums of Integer Functions</a>, J. Integer Sequences, Vol. 14 (2011), Article 11.9.1.
%H Kival Ngaokrajang, <a href="/A002620/a002620.pdf">Illustration of twin hearts patterns (6c4c): T, U, V</a>
%H Brian O'Sullivan and Thomas Busch, <a href="http://arxiv.org/abs/0810.0231">Spontaneous emission in ultracold spinpolarised anisotropic Fermi seas</a>, arXiv:0810.0231v1 [quantph], 2008. [Eq 8a, lambda=2]
%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992.
%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Université du Québec à Montréal, 1992.
%H N. Reading, <a href="http://www4.ncsu.edu/~nreadin/papers/dissective.pdf">Order Dimension, Strong Bruhat Order and Lattice Properties for Posets </a>
%H N. Reading, <a href="http://doi.org/10.1023/A:1015287106470">Order Dimension, Strong Bruhat Order and Lattice Properties for Posets</a>, Order, Vol. 19, no. 1 (2002), 73100.
%H J. Scholes, <a href="https://mks.mff.cuni.cz/kalva/putnam/putn66.html">27th Putnam 1966 Prob. A1</a>
%H N. J. A. Sloane, <a href="/classic.html#LOSS">Classic Sequences</a>
%H Sam E. Speed, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Speed/speed11.html">The Integer Sequence A002620 and Upper Antagonistic Functions</a>, Journal of Integer Sequences, Vol. 6 (2003), Article 03.1.4
%H K. E. Stange, <a href="http://arxiv.org/abs/1108.3051">Integral points on elliptic curves and explicit valuations of division polynomials</a> arXiv:1108.3051v3 [math.NT], 20112014.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BlackBishopGraph.html">Black Bishop Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MatchingNumber.html">Matching Number</a>
%H Thomas Wieder, The number of certain kcombinations of an nset, <a href="http://www.math.nthu.edu.tw/~amen/2008/070301.pdf">Applied Mathematics Electronic Notes</a>, vol. 8 (2008).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bisymmetric_matrix">Bisymmetric Matrix</a>.
%H <a href="/index/Tu#2wis">Index entries for twoway infinite sequences</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,2,1).
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F a(n) = (2*n^21+(1)^(n))/8.  _Paul Barry_, May 27 2003
%F G.f.: x^2/((1x)^2*(1x^2)).
%F E.g.f.: exp(x)*(2*x^2+2*x1)/8 + exp(x)/8.
%F a(n) = 2*a(n1)  2*a(n3) + a(n4).  _Jaume Oliver Lafont_, Dec 05 2008
%F a(n) = a(n) for all n in Z.
%F a(n) = a(n1) + floor(n/2), n > 0. Partial sums of A004526.  _Adam Kertesz_, Sep 20 2000
%F a(n) = a(n1) + a(n2)  a(n3) + 1 [with a(1) = a(0) = a(1) = 0], a(2k) = k^2, a(2k1) = k(k1).  _Henry Bottomley_, Mar 08 2000
%F 0*0, 0*1, 1*1, 1*2, 2*2, 2*3, 3*3, 3*4, ... with an obvious pattern.
%F a(n) = Sum_{k=1..n} floor(k/2).  Yong Kong (ykong(AT)curagen.com), Mar 10 2001
%F a(n) = n*floor((n1)/2)  floor((n1)/2)*(floor((n1)/2)+ 1); a(n) = a(n2) + n2 with a(1) = 0, a(2) = 0.  _Santi Spadaro_, Jul 13 2001
%F Also: a(n) = binomial(n, 2)  a(n1) = A000217(n1)  a(n1) with a(0) = 0.  _Labos Elemer_, Apr 26 2003
%F a(n) = Sum_{k=0..n} (1)^(nk)*C(k, 2).  _Paul Barry_, Jul 01 2003
%F a(n) = (1)^n * partial sum of alternating triangular numbers.  _Jon Perry_, Dec 30 2003
%F a(n) = A024206(n+1)  n.  _Philippe Deléham_, Feb 27 2004
%F a(n) = a(n2) + n  1, n > 1.  _Paul Barry_, Jul 14 2004
%F a(n+1) = Sum_{i=0..n} min(i, ni).  _Marc LeBrun_, Feb 15 2005
%F a(n+1) = Sum_{k = 0..floor((n1)/2)} n2k; a(n+1) = Sum_{k=0..n} k*(1(1)^(n+k1))/2.  _Paul Barry_, Apr 16 2005
%F a(n) = A108561(n+1,n2) for n > 2.  _Reinhard Zumkeller_, Jun 10 2005
%F 1 + 1/(1 + 2/(1 + 4/(1 + 6/(1 + 9/(1 + 12/(1 + 16/(1 + . . ))))))) = 6/(Pi^2  6) = 1.550546096730...  _Philippe Deléham_, Jun 20 2005
%F For n > 2 a(n) = a(n1) + ceiling(sqrt(a(n1))).  _Jonathan Vos Post_, Jan 19 2006
%F Sequence starting (2, 2, 4, 6, 9, ...) = A128174 (as an infinite lower triangular matrix) * vector [1, 2, 3, ...]; where A128174 = (1; 0,1; 1,0,1; 0,1,0,1; ...).  _Gary W. Adamson_, Jul 27 2007
%F a(n) = Sum_{i=k..n} P(i, k) where P(i, k) is the number of partitions of i into k parts.  _Thomas Wieder_, Sep 01 2007
%F a(n) = sum of row (n2) of triangle A115514.  _Gary W. Adamson_, Oct 25 2007
%F For n > 1: gcd(a(n+1), a(n)) = a(n+1)  a(n).  _Reinhard Zumkeller_, Apr 06 2008
%F a(n+3) = a(n) + A000027(n) + A008619(n+1) = a(n) + A001651(n+1) with a(1) = 0, a(2) = 0, a(3) = 1.  _Yosu Yurramendi_, Aug 10 2008
%F a(2n) = A000290(n). a(2n+1) = A002378(n).  _Gary W. Adamson_, Nov 29 2008
%F a(n+1) = a(n) + A110654(n).  _Reinhard Zumkeller_, Aug 06 2009
%F a(n) = Sum_{k=0..n} (k mod 2)*(nk); Cf. A000035, A001477.  _Reinhard Zumkeller_, Nov 05 2009
%F a(n1) = (n*n  2*n + n mod 2)/4.  _Ctibor O. Zizka_, Nov 23 2009
%F a(n) = round((2*n^21)/8) = round(n^2/4) = ceiling((n^21)/4).  _Mircea Merca_, Nov 29 2010
%F n*a(n+2) = 2*a(n+1) + (n+2)*a(n). Holonomic Ansatz with smallest order of recurrence.  _Thotsaporn Thanatipanonda_, Dec 12 2010
%F a(n+1) = (n*(2+n) + n mod 2)/4.  _Fred Daniel Kline_, Sep 11 2011
%F a(n) = A199332(n, floor((n+1)/2)).  _Reinhard Zumkeller_, Nov 23 2011
%F a(n) = floor(b(n)) with b(n) = b(n1) + n/(1+e^(1/n)) and b(0)= 0.  _Richard R. Forberg_, Jun 08 2013
%F a(n) = Sum_{i=1..floor((n+1)/2)} (n+1)2i.  _Wesley Ivan Hurt_, Jun 09 2013
%F a(n) = floor((n+2)/2  1)*(floor((n+2)/2)1 + (n+2) mod 2).  _Wesley Ivan Hurt_, Jun 09 2013
%F Sum_{n>=2} 1/a(n) = 1 + Zeta(2) = 1+A013661.  _Enrique Pérez Herrero_, Jun 30 2013
%F Empirical: a(n) = floor(n/(e^(4/n)1).  _Richard R. Forberg_, Jul 24 2013
%F a(n) = A007590(n)/2.  _Wesley Ivan Hurt_, Mar 08 2014
%F A240025(a(n)) = 1.  _Reinhard Zumkeller_, Jul 05 2014
%F 0 = a(n)*a(n+2) + a(n+1)*(2*a(n+2) + a(n+3)) for all integers n.  _Michael Somos_, Nov 22 2014
%F a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+jn1)/2).  _Wesley Ivan Hurt_, Mar 12 2015
%F a(4n+1) = A002943(n) for all n>=0.  _M. F. Hasler_, Oct 11 2015
%F a(n+2)a(n2) = A004275(n+1).  _Anton Zakharov_, May 11 2017
%F a(n) = floor(n/2)*floor((n+1)/2).  _Bruno Berselli_, Jun 08 2017
%e a(3) = 2, floor(3/2)*ceiling(3/2) = 2.
%e [ n] a(n)
%e 
%e [ 2] 1
%e [ 3] 2
%e [ 4] 1 + 3
%e [ 5] 2 + 4
%e [ 6] 1 + 3 + 5
%e [ 7] 2 + 4 + 6
%e [ 8] 1 + 3 + 5 + 7
%e [ 9] 2 + 4 + 6 + 8
%e From _Wolfdieter Lang_, Dec 09 2014: (Start)
%e Tiling of a triangular shape T_N, N >= 1 with rectangles:
%e N=5, n=6: a(6) = 9 because all the rectangles (i, j) (modulo transposition, i.e., interchange of i and j) which are of use are:
%e (5, 1) ; (1, 1)
%e (4, 2), (4, 1) ; (2, 2), (2, 1)
%e ; (3, 3), (3, 2), (3, 1)
%e That is (1+1) + (2+2) + 3 = 9 = a(6). Partial sums of 1, 1, 2, 2, 3, ... (A004526). (End)
%e Bisymmetric matrices B: 2 X 2, a(3) = 2 from B[1,1] and B[1,2]. 3 X 3, a(4) = 4 from B[1,1], B[1,2], B[1,3], and B[2,2].  _Wolfdieter Lang_, Jul 07 2015
%e From _John M. Campbell_, Jan 29 2016: (Start)
%e Letting n=5, there are a(n)=a(5)=6 partitions of 2n+1=11 of length three with exactly two even entries:
%e (8,2,1)  2n+1
%e (7,2,2)  2n+1
%e (6,4,1)  2n+1
%e (6,3,2)  2n+1
%e (5,4,2)  2n+1
%e (4,4,3)  2n+1
%e (End)
%p A002620 := n>floor(n^2/4); G002620 := series(x^2/((1x)^2*(1x^2)),x,60);
%p with(combstruct):ZL:=[st,{st=Prod(left,right),left=Set(U,card=r),right=Set(U,card<r),U=Sequence(Z,card>=1)}, unlabeled]: subs(r=1,stack): seq(count(subs(r=2,ZL),size=m),m=0..57) ; # _Zerinvary Lajos_, Mar 09 2007
%p A002620:=1/(z+1)/(z1)^3; # _Simon Plouffe_ in his 1992 dissertation, leading zeros dropped
%p A002620 := n > add(k, k = select(k > k mod 2 <> n mod 2, [$1 .. n])): seq(A002620(n), n = 0 .. 57);
%p # _Peter Luschny_, Jul 06 2011
%t a[n_] := Ceiling[n/2]Floor[n/2]; Table[ a[n], {n, 0, 56}] (* _Robert G. Wilson v_, Jun 18 2005 *)
%t a = 0; Table[(a = n^2 + n  a)/2, {n, 1, 90}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 18 2009 *)
%t a[n_] := a[n] = 2a[n  1]  2a[n  3] + a[n  4]; a[0] = a[1] = 0; a[2] = 1; a[3] = 2; Array[a, 60, 0] (* _Robert G. Wilson v_, Mar 28 2011 *)
%t LinearRecurrence[{2, 0, 2, 1}, {0, 0, 1, 2}, 60] (* _Harvey P. Dale_, Oct 05 2012 *)
%t f[n_] := Block[{c = 0, m = n+1}, Do[ If[ MemberQ[ Range[x, y], (x + y)/2], c++ ], {x, m  1}, {y, x + 1, m}]; c] (* _Robert G. Wilson v_, May 22 2014 *)
%o (MAGMA) [ Floor(n/2)*Ceiling(n/2) : n in [0..40]];
%o (PARI) a(n)=n^2\4
%o (PARI) t(n)=n*(n+1)/2 for(i=1,50,print1(","(1)^i*sum(k=1,i,(1)^k*t(k))))
%o (PARI) a(n)=n^2>>2 \\ _Charles R Greathouse IV_, Nov 11 2009
%o (PARI) x='x+O('x^100); concat([0, 0], Vec(x^2/((1x)^2*(1x^2)))) \\ _Altug Alkan_, Oct 15 2015
%o (Haskell)
%o a002620 = (`div` 4) . (^ 2)  _Reinhard Zumkeller_, Feb 24 2012
%o (Maxima) makelist(floor(n^2/4),n,0,50); /* _Martin Ettl_, Oct 17 2012 */
%o (Sage)
%o def A002620():
%o x, y = 0, 1
%o yield x
%o while true:
%o yield x
%o x, y = x + y, x//y + 1
%o a = A002620(); print [a.next() for i in range(58)] # _Peter Luschny_, Dec 17 2015
%o (GAP) # using the formula by Paul Barry
%o A002620 := List([1..10^4], n> (2*n^2  1 + (1)^n)/8); # _Muniru A Asiru_, Feb 01 2018
%Y A087811 is another version of this sequence.
%Y Cf. A024206, A072280, A002984, A007590, A000212, A118015, A056827, A118013, A128174, A000601, A115514, A189151, A063657, A171608, A005044, A030179, A275437, A004526.
%Y Differences of A002623. Complement of A049068.
%Y a(n) = A014616(n2) + 2 = A033638(n)  1 = A078126(n) + 1. Cf. A055802, A055803.
%Y Antidiagonal sums of array A003983.
%Y Cf. A033436  A033444.  _Reinhard Zumkeller_, Nov 30 2009
%Y Cf. A008233, A008217, A014980, A197081, A197122.
%Y Elliptic troublemaker sequences: A000212 (= R_n(1,3) = R_n(2,3)), A007590 (= R_n(2,4)), A030511 (= R_n(2,6) = R_n(4,6))), A033436 (= R_n(1,4) = R_n(3,4)), A033437 (= R_n(1,5) = R_n(4,5)), A033438 (= R_n(1,6) = R_n(5,6)), A033439 (= R_n(1,7) = R_n(6,7)), A184535 (= R_n(2,5) = R_n(3,5)).
%Y Cf. A077043, A060656 (2^a(n)).
%K nonn,easy,nice,core,changed
%O 0,4
%A _N. J. A. Sloane_
