|
%I M0998 N0374 #797 Mar 22 2024 08:33:45
%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 Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4).
%C b(n) = a(n+2) is the number of multigraphs with loops on 2 nodes with n edges [so g.f. for b(n) is 1/((1-x)^2*(1-x^2))]. Also number of 2-covers of an n-set; 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 triangle-free 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_{n-1} (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, non-capturing 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 non-symmetric 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(n-1) gives number of distinct elements greater than 1 of non-symmetric 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_, 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/(1-x^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 n-th 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+1) is the number of noncongruent integer-sided triangles with largest side n. - _David W. Wilson_ [Comment corrected Sep 26 2006]
%C A quarter-square table can be used to multiply integers since n*m = a(n+m) - a(n-m) 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(n-1) 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 2-colored 3-partition of n. - _Ctibor O. Zizka_, Nov 19 2014
%C Also a(n-1) = 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 even-indexed 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 2-dimensional irrep, along with any combination of the two 1-dimensional 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 5-repetitive 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(n-1) + ceiling (a(n-1)) this is to ceiling as A002984 is to floor, and as A033638 is to round. - _Jonathan Vos Post_, Oct 08 2011
%C a(n-2) gives the number of distinct graphs with n vertices and n regions. - _Erik Hasse_, Oct 18 2011
%C Construct the n-th 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 n-element 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 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 = 1-k 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 quarter-squares; 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 01-avoiding 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 01-avoiding 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 n-vertex triangle-free 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
%C Maximum area of a rectangle with perimeter 2n and sides of integer length. - _André Engels_, Jul 29 2018
%C Also the crossing number of the complete bipartite graph K_{3,n+1}. - _Eric W. Weisstein_, Sep 11 2018
%C a(n+2) is the number of distinct genotype frequency vectors possible for a sample of n diploid individuals at a biallelic genetic locus with a specified major allele. Such vectors are the lists of nonnegative genotype frequencies (n_AA, n_AB, n_BB) with n_AA + n_AB + n_BB = n and n_AA >= n_BB. - _Noah A Rosenberg_, Feb 05 2019
%C a(n+2) is the number of distinct real spectra (eigenvalues repeated according to their multiplicity) for an orthogonal n X n matrix. The case of an empty spectrum list is logically counted as one of those possibilities, when it exists. Thus a(n+2) is the number of distinct reduced forms (on the real field, in orthonormal basis) for elements in O(n). - _Christian Devanz_, Feb 13 2019
%C a(n) is the number of non-isomorphic asymmetric graphs that can be created by adding a single edge to a path on n+4 vertices. - Emma Farnsworth, Natalie Gomez, Herlandt Lino, and _Darren Narayan_, Jul 03 2019
%C a(n+1) is the number of integer triangles with maximum side-length n. - _James East_, Oct 30 2019
%C a(n) is the number of nonempty subsets of {1,2,...,n} that contain exactly one odd and one even number. For example, for n=7, a(7)=12 and the 12 subsets are {1,2}, {1,4}, {1,6}, {2,3}, {2,5}, {2,7}, {3,4}, {3,6}, {4,5}, {4,7}, {5,6}, {6,7}. - _Enrique Navarrete_, Dec 16 2019
%C a(n+1) is also the n-th term of the Saind sequence (w_n)_{n>=1}, i.e., the infinite sequence caused by the entries of the queue of the degree sequences associated with the Saind arrays, as n increases. - _Giulia Palma_, Jun 24 2020
%C Aside from the first two terms, a(n) enumerates the number of distinct normal ordered terms in the expansion of the differential operator (x + d/dx)^m associated to the Hermite polynomials and the Heisenberg-Weyl algebra. It also enumerates the number of distinct monomials in the bivariate polynomials corresponding to the partial sums of the series for cos(x+y) and sin(x+y). Cf. A344678. - _Tom Copeland_, May 27 2021
%C a(n) is the maximal number of negative products a_i * a_j (1 <= i <= j <= n), where all a_i are real numbers. - _Logan Pipes_, Jul 08 2021
%C From _Allan Bickle_, Dec 20 2021: (Start)
%C a(n) is the maximum product of the chromatic numbers of a graph of order n-1 and its complement. The extremal graphs are characterized in the papers of Finck (1968) and Bickle (2023).
%C a(n) is the maximum product of the degeneracies of a graph of order n+1 and its complement. The extremal graphs are characterized in the paper of Bickle (2012). (End)
%C a(n) is the maximum number m such that m white rooks and m black rooks can coexist on an n-1 X n-1 chessboard without attacking each other. - _Aaron Khan_, Jul 13 2022
%C Partial sums of A004526. - _Bernard Schott_, Jan 06 2023
%C a(n) is the number of 231-avoiding odd Grassmannian permutations of size n. - _Juan B. Gil_, Mar 10 2023
%C a(n) is the number of integer tuples (x,y) satisfying n + x + y >= 0, 25*n + x - 11*y >=0, 25*n - 11*x + y >=0, n + x + y == 0 (mod 12) , 25*n + x - 11*y == 0 (mod 5), 25*n - 11*x + y == 0 (mod 5) . For n=2, the sole solution is (x,y) = (0,0) and so a(2) = 1. For n = 3, the a(3) = 2 solutions are (-3, 2) and (2, -3). - _Jeffery Opoku_, Feb 16 2024
%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: 1965-1984, M.A.A., 1985; see Problem A-1 of 27th Competition.
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 25.
%D Michael Doob, The Canadian Mathematical Olympiad -- L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society -- Société Mathématique du Canada, Problème 9, 1970, pp 22-23, 1993.
%D H. J. Finck, On the chromatic numbers of a graph and its complement. Theory of Graphs (Proc. Colloq., Tihany, 1966) Academic Press, New York (1968), 99-113.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 99.
%D D. E. Knuth, The art of programming, Vol. 1, 3rd Edition, Addison-Wesley, 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. Adams-Watters, <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 BP-XOR Codes</a>, arXiv:1709.07949 [cs.IT], 2017.
%H J. A. Bate and 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) 73-83.
%H M. Benoumhani and M. Kolli, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Benoumhani/benoumhani6.html">Finite topologies and partitions</a>, JIS, Vol. 13 (2010), Article 10.3.5, Lemma 6 first line.
%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/ngcorepaper.pdf">Nordhaus-Gaddum Theorems for k-Decompositions</a>, Congr. Num., Vol. 211 (2012), pp. 171-183.
%H Allan Bickle, <a href="https://doi.org/10.1016/j.disc.2023.113392">Extremal Decompositions for Nordhaus-Gaddum Theorems</a>, Discrete Math, 346 7 (2023), 113392.
%H G. Blom and C.-E. Froeberg, <a href="/A002575/a002575.pdf">Om myntvaexling (On money-changing) [Swedish]</a>, Nordisk Matematisk Tidskrift, Vol. 10 (1962), pp. 55-69, 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 Henry 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 n-cube</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 and A Cihangir, <a href="http://arxiv.org/abs/1512.03044">Enumeration and investigation of acute 0/1-simplices 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., Vol. 167/168 (1997), pp. 605-615.
%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. Seq., Vol. 3 (2000), Article 00.1.5.
%H Johann Cigler, <a href="https://arxiv.org/abs/1711.03340">Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle</a>, arXiv:1711.03340 [math.CO], 2017.
%H F. Javier de Vega, <a href="https://arxiv.org/abs/2003.13378">An extension of Furstenberg's theorem of the infinitude of primes</a>, arXiv:2003.13378 [math.NT], 2020.
%H Bakir Farhi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Farhi/farhi7.html">On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a)</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.4.
%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., Vol. 26 (1955), pp. 301-312.
%H E. Fix and J. L. Hodges, <a href="/A000601/a000601.pdf">Significance probabilities of the Wilcoxon test</a>, Annals Math. Stat., Vol. 26 (1955), pp. 301-312. [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 Juan B. Gil and Jessica A. Tomasko, <a href="https://arxiv.org/abs/2207.12617">Pattern-avoiding even and odd Grassmannian permutations</a>, arXiv:2207.12617 [math.CO], 2022.
%H J. W. L. Glaisher and G. Carey Foster, <a href="https://www.jstor.org/stable/41135937">The Method of Quarter Squares</a>, Journal of the Institute of Actuaries, Vol. 28, No. 3, January, 1890, pp. 227-235.
%H E. Gottlieb and M. Sheard, <a href="http://discretews.math.msstate.edu/2014/gottlieb_slides.pdf">An Erdős-Szekeres 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, June-August 1968</a>
%H Phillip Tomas Heikoop, <a href="https://digitalcommons.wpi.edu/mqp-all/6822/">Dimensions of Matrix Subalgebras</a>, Bachelor's Thesis, Worcester Polytechnic Institute (Massachusetts, 2019).
%H Andreas M. Hinz and Paul K. Stockmeyer, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Hinz/hinz5.html">Precious Metal Sequences and Sierpinski-Type Graphs</a>, J. Integer Seq., Vol 25 (2022), Article 22.4.8.
%H T. Huber, N. Mayes, J. Opoku, and D. Ye, <a href="https://doi.org/10.1016/j.jnt.2023.11.009">Ramanujan type congruences for quotients of Klein forms</a>, Journal of Number Theory, 258, 281-333, (2024). See Corollary 2.5 page 11.
%H Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 36.
%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), 881-888. 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, Vol. 107 (Aug. 2000), pp. 634-639.
%H Paloma Jiménez-Sepúlveda and Luis Manuel Rivera, <a href="https://arxiv.org/abs/1810.06354">Independence numbers of some double vertex graphs and pair graphs</a>, arXiv:1810.06354 [math.CO], 2018.
%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 Seq., Vol. 7 (2004), Article 04.1.6.
%H Jukka Kohonen, <a href="https://arxiv.org/abs/1804.03679">Counting graded lattices of rank three that have few coatoms</a>, arXiv:1804.03679 [math.CO], 2018.
%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 and P.-L. Nguyen, <a href="https://lirias.kuleuven.be/1656296">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 Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber., Vol. 30 (1897), pp. 1917-1926.
%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber., Vol. 30 (1897), pp. 1917-1926. (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, Vol. 10 (1907), pp. 60-61.
%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 Emanuele Munarini, <a href="https://doi.org/10.4418/2021.76.1.14">Topological indices for the antiregular graphs</a>, Le Mathematiche, Vol. 76, No. 1 (2021), pp. 277-310, see page 282.
%H Kival Ngaokrajang, <a href="/A002620/a002620.pdf">Illustration of twin hearts patterns (6c4c): T, U, V</a>.
%H E. A. Nordhaus and J. Gaddum, <a href="https://lost-contact.mit.edu/afs//nada.kth.se/home/p/u1wdw8tp/Downloads/2306658.pdf">On complementary graphs</a>, Amer. Math. Monthly, Vol. 63 (1956), pp. 175-177.
%H Brian O'Sullivan and Thomas Busch, <a href="http://arxiv.org/abs/0810.0231">Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas</a>, arXiv:0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=2]
%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 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), pp. 73-100.
%H Denis Roegel, <a href="https://locomat.loria.fr/blater1887/blater1887doc.pdf">A reconstruction of Blater’s table of quarter-squares (1887)</a>, Locomat Project, 6 November 2013.
%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], 2011-2014.
%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/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCrossingNumber.html">Graph Crossing Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MatchingNumber.html">Matching Number</a>.
%H Thomas Wieder, <a href="http://www.math.nthu.edu.tw/~amen/2008/070301.pdf">The number of certain k-combinations of an n-set</a>, Applied Mathematics Electronic Notes, Vol. 8 (2008), pp. 45-52.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bisymmetric_matrix">Bisymmetric Matrix</a>.
%H <a href="/index/Tu#2wis">Index entries for two-way 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^2-1+(-1)^n)/8. - _Paul Barry_, May 27 2003
%F G.f.: x^2/((1-x)^2*(1-x^2)) = x^2 / ( (1+x)*(1-x)^3 ). - _Simon Plouffe_ in his 1992 dissertation, leading zeros dropped
%F E.g.f.: exp(x)*(2*x^2+2*x-1)/8 + exp(-x)/8.
%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - _Jaume Oliver Lafont_, Dec 05 2008
%F a(-n) = a(n) for all n in Z.
%F a(n) = a(n-1) + floor(n/2), n > 0. Partial sums of A004526. - _Adam Kertesz_, Sep 20 2000
%F a(n) = a(n-1) + a(n-2) - a(n-3) + 1 [with a(-1) = a(0) = a(1) = 0], a(2k) = k^2, a(2k-1) = k(k-1). - _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((n-1)/2) - floor((n-1)/2)*(floor((n-1)/2)+ 1); a(n) = a(n-2) + n-2 with a(1) = 0, a(2) = 0. - _Santi Spadaro_, Jul 13 2001
%F Also: a(n) = binomial(n, 2) - a(n-1) = A000217(n-1) - a(n-1) with a(0) = 0. - _Labos Elemer_, Apr 26 2003
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*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(n-2) + n - 1, n > 1. - _Paul Barry_, Jul 14 2004
%F a(n+1) = Sum_{i=0..n} min(i, n-i). - _Marc LeBrun_, Feb 15 2005
%F a(n+1) = Sum_{k = 0..floor((n-1)/2)} n-2k; a(n+1) = Sum_{k=0..n} k*(1-(-1)^(n+k-1))/2. - _Paul Barry_, Apr 16 2005
%F a(n) = A108561(n+1,n-2) 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 a(n) = Sum_{k=0..n} Min_{k, n-k}, sums of rows of the triangle in A004197. - _Reinhard Zumkeller_, Jul 27 2005
%F For n > 2 a(n) = a(n-1) + ceiling(sqrt(a(n-1))). - _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 (n-2) 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)*(n-k); Cf. A000035, A001477. - _Reinhard Zumkeller_, Nov 05 2009
%F a(n-1) = (n*n - 2*n + n mod 2)/4. - _Ctibor O. Zizka_, Nov 23 2009
%F a(n) = round((2*n^2-1)/8) = round(n^2/4) = ceiling((n^2-1)/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(n-1) + 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-1) = 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 A237347(a(n)) = 3; A235711(n) = A003415(a(n)). - _Reinhard Zumkeller_, Mar 18 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+j-n-1)/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(n-2) = A004275(n+1). - _Anton Zakharov_, May 11 2017
%F a(n) = floor(n/2)*floor((n+1)/2). - _Bruno Berselli_, Jun 08 2017
%F a(n) = a(n-3) + floor(3*n/2) - 2. - _Yuchun Ji_, Aug 14 2020
%F a(n)+a(n+1) = A000217(n). - _R. J. Mathar_, Mar 13 2021
%F a(n) = A004247(n,floor(n/2)). - _Logan Pipes_, Jul 08 2021
%F a(n) = floor(n^2/2)/2. - _Clark Kimberling_, Dec 05 2021
%F Sum_{n>=2} (-1)^n/a(n) = Pi^2/6 - 1. - _Amiram Eldar_, Mar 10 2022
%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)
%e From _Aaron Khan_, Jul 13 2022: (Start)
%e Examples of the sequence when used for rooks on a chessboard:
%e .
%e A solution illustrating a(5)=4:
%e +---------+
%e | B B . . |
%e | B B . . |
%e | . . W W |
%e | . . W W |
%e +---------+
%e .
%e A solution illustrating a(6)=6:
%e +-----------+
%e | B B . . . |
%e | B B . . . |
%e | B B . . . |
%e | . . W W W |
%e | . . W W W |
%e +-----------+
%e (End)
%p A002620 := n->floor(n^2/4); G002620 := series(x^2/((1-x)^2*(1-x^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
%t Table[Ceiling[n/2] Floor[n/2], {n, 0, 56}] (* _Robert G. Wilson v_, Jun 18 2005 *)
%t LinearRecurrence[{2, 0, -2, 1}, {0, 0, 1, 2}, 60] (* _Harvey P. Dale_, Oct 05 2012 *)
%t Table[Floor[n^2/4], {n, 0, 20}] (* _Eric W. Weisstein_, Sep 11 2018 *)
%t Floor[Range[0, 20]^2/4] (* _Eric W. Weisstein_, Sep 11 2018 *)
%t CoefficientList[Series[-(x^2/((-1 + x)^3 (1 + x))), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 11 2018 *)
%t Table[Floor[n^2/2]/2, {n, 0, 56}] (* _Clark Kimberling_, Dec 05 2021 *)
%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/((1-x)^2*(1-x^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([next(a) 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
%o (Python)
%o def A002620(n): return (n**2)>>2 # _Chai Wah Wu_, Jul 07 2022
%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(n-2) + 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. 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)), A344678.
%Y Cf. A000290, A056834, A130519, A056838, A056865.
%Y Cf. A250000 (queens on a chessboard), A176222 (kings on a chessboard), A355509 (knights on a chessboard).
%Y Maximal product of k positive integers with sum n, for k = 2..10: this sequence (k=2), A006501 (k=3), A008233 (k=4), A008382 (k=5), A008881 (k=6), A009641 (k=7), A009694 (k=8), A009714 (k=9), A354600 (k=10).
%K nonn,easy,nice,core
%O 0,4
%A _N. J. A. Sloane_
|
