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 A002620 Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4). (Formerly M0998 N0374) 212

%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 Quarter-squares: 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/((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 (spados(AT)katamail.com), 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 denotes 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 (amarnath_murthy(AT)yahoo.com), 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 (amarnath_murthy(AT)yahoo.com), 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 (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 (zbi74583(AT)boat.zero.ad.jp), 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. Partial sums of A004526 (nonnegative integers repeated: partitions into two parts). - _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 a(n) = A108561(n+1,n-2) for n > 2. - _Reinhard Zumkeller_, Jun 10 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 Christams" song. For example, on the fifth day of Christmas, you have 9 French hens. - _Alonso del Arte_, Jun 17 2005

%C a(n) = Sum(Min{k, n-k}: 0 <= k <= n), sums of rows of the triangle in A004197. - _Reinhard Zumkeller_, Jul 27 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 a(n+1) = a(n) + A110654(n). [_Reinhard Zumkeller_, Aug 06 2009]

%C a(n) = (n*n - 2*n + n mod 2)/4 [Ctibor O. Zizka, Nov 23 2009]

%C Also a(n) = 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 all the even-indexed terms are the average of their 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) counts 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 doublefactorial. - _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 8, 2011].

%C a(n-2) counts 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

%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

%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 27-th Competition.

%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 25.

%D J. A. Bate & G. H. J. van Rees, The Size of the Smallest Strong Critical Set in a Latin Square, Ars Combinatoria, Vol. 53 (1999) 73-83.

%D E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.

%D A. Ganesan, Automorphism groups of graphs, Arxiv preprint arXiv:1206.6279, 2012. - From _N. J. A. Sloane_, Dec 17 2012

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 99.

%D O. A. Ivanov, On the number of regions into which n straight lines divide the plane, Amer. Math. Monthly, 117 (2010), 881-888. See Th. 4.

%D T. Jenkyns and E. Muller, Triangular triples from ceilings to floors, Amer. Math. Monthly, 107 (Aug. 2000), 634-639.

%D D. E. Knuth, The art of programming, Vol. 1, 3rd Edition, Addison-Wesley, 1997, Ex. 36 of section 1.2.4.

%D S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

%D Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions, J. Integer Seq., 14(2011), Article 11.9.1.

%D J. Nelder, Critical sets in Latin squares, CSIRO Division of Math. and Stats. Newsletter, Vol. 38 (1977), p. 4.

%D N. Reading, Order Dimension, Strong Bruhat Order and Lattice Properties for Posets, Order, Vol. 19, no. 1 (2002), 73-100.

%D Brian OSullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 8a, lambda=2]

%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 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 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), 605-615.

%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H INRIA Algorithms Project, <a href="http://algo.inria.fr/ecs/ecs?searchType=1&amp;service=Search&amp;searchTerms=105">Encyclopedia of Combinatorial Structures 105</a>

%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/index.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>

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">1031 Generating Functions and Conjectures</a>, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

%H N. Reading, <a href="http://www.math.umn.edu/~reading/dissective.ps">Order Dimension, Strong Bruhat Order and Lattice Properties for Posets </a>

%H J. Scholes, <a href="http://www.kalva.demon.co.uk/putnam/psoln/psol661.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/index.html">"The Integer Sequence A002620 and Upper Antagonistic Functions" </a>, Journal of Integer Sequences, Vol. 5 (2002), Article 03.1.4

%H Thomas Wieder, The number of certain k-combinations of an n-set, <a href="http://www.math.nthu.edu.tw/~amen/">Applied Mathematics Electronic Notes</a>, vol. 8 (2008).

%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>

%H <a href="/index/Rea#recLCC">Index entries for sequences related to linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

%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)).

%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).

%F a(n) = a(n-1) + int(n/2), n > 0. - Adam Kertesz (adamkertesz(AT)worldnet.att.net), 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(floor(k/2), k = 1..n). - 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 (spados(AT)katamail.com), Jul 13 2001

%F Also: a(n) = C(n, 2) - a(n-1) = A000217(n-1) - a(n-1) with a(0) = 0. - Labos E. (labos(AT)ana.hu), 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 DelĂ©ham_, Feb 27 2004

%F Partial sums of A004526. - _Lekraj Beedassy_, Jun 30 2004

%F a(n) = a(n-2) + n - 1, a(0) = 0, a(1) = 0. - _Paul Barry_, Jul 14 2004

%F a(n+1) = sum min(i, n-i), i = 0..n. - _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 1 + 1/(1 + 2/(1 + 4/(1 + 6/(1 + 9/(1 + 12/(1 + 16/(1 + . . ))))))) = 6/(Pi^2 - 6) = 1.550546096730... - _Philippe DELEHAM_, Jun 20 2005

%F a(0) = 0; a(1) = 0; a(2) = 1; 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 finite 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), a(3)=1. - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Aug 10 2008

%F a(n) = SUM((k mod 2)*(n-k): 0 <= k <= n), cf. A000035, A001477. [_Reinhard Zumkeller_, Nov 05 2009]

%F a(n) = round((2*n^2-1)/8) = round(n^2/4) = ceil((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) = (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 G.f.: G(0)*x^2/(2*(1-x^2)*(1-x)), where G(k)= 1 + 1/(1 - x*(2*k+1)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 25 2013

%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

%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

%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 (zerinvarylajos(AT)yahoo.com), Mar 09 2007

%p A002620:=-1/(z+1)/(z-1)^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 f[n_] := Ceiling[n/2]Floor[n/2]; Table[ f[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_, March 28 2011 *)

%t LinearRecurrence[{2, 0, -2, 1}, {0, 0, 1, 2}, 60] (* _Harvey P. Dale_, Oct 05 2012 *)

%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 a002620 = (`div` 4) . (^ 2) -- _Reinhard Zumkeller_, Feb 24 2012

%o (Maxima) makelist(floor(n^2/4),n,0,50); /* _Martin Ettl_, Oct 17 2012 */

%Y A087811 is another version of this sequence.

%Y Cf. A024206, A072280, A002984, A007590, A000212, A118015, A056827, A118013, A128174, A000601, A115514, A189151, A063657, A171608, A007590, A005044, A030179.First differences give integers repeated (cf. A008619 or 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 a(2n) = A000290(n) = squares, a(2n+1)=A002378(n) = oblong numbers. A122196 [From Gary W. Adamson, Nov 29 2008]

%Y Cf. A033436, A033437, A033438, A033439, A033440, A033441, A033442, A033443, A033444. [From _Reinhard Zumkeller_, Nov 30 2009]

%Y Cf. A008233, A008217, A014980, A197081, A197122.

%K nonn,easy,nice,core,changed

%O 0,4

%A _N. J. A. Sloane_.

%E Removed attribute "conjectured" from _Simon Plouffe_ g.f., _R. J. Mathar_, Mar 11 2009

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