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A002620
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Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).
(Formerly M0998 N0374)
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173
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0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 121, 132, 144, 156, 169, 182, 196, 210, 225, 240, 256, 272, 289, 306, 324, 342, 361, 380, 400, 420, 441, 462, 484, 506, 529, 552, 576, 600, 625, 650, 676, 702, 729, 756, 784, 812
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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 (vladeta(AT)eunet.rs), Jun 08, 2000.
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
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
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
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 (benoit7848c(AT)orange.fr), Jun 19 2002
Sums of pairs of neighboring terms are triangular numbers in increasing order. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 19 2002
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
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
Terms are the geometric mean and arithmetic mean of their neighbors alternately. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 17 2002
Maximum product of two integers whose sum is n. - Matthew Vandermast, Mar 04 2003
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
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
Union of square numbers (A000290) and oblong numbers (A002378). - Lekraj Beedassy, Oct 02 2003
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
a(n) gives number of maximal strokes on complete graph K_n, when edges on K_n can be asigned 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
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
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 Shepherd, Feb 27 2004
See A092186 for another application.
Also, the number of nonisomorphic transversal combinatorial geometries of rank 2. - Alexandr S. Radionov (rasmailru(AT)mail.ru), Jun 02 2004
a(n+1) is the transform of n under the Riordan array (1/(1-x^2),x). - Paul Barry, Apr 16 2005
a(n) = A108561(n+1,n-2) for n>2. - Reinhard Zumkeller, Jun 10 2005
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. - Alonso Del Arte, Jun 17 2005
a(n) = Sum(Min{k,n-k}: 0<=k<=n), sums of rows of the triangle in A004197. - Reinhard Zumkeller, Jul 27 2005
a(n+1) is the number of noncongruent integer-sided triangles with largest side n - David W. Wilson. [Comment corrected Sep 26 2006]
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
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
Maximal number of squares (maximal area) in a polyomino with perimeter 2n. - Tanya Khovanova, Jul 04 2007
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 (webonfim(AT)bol.com.br), Jul 26 2008
Equals row sums of triangle A122196 [From Gary W. Adamson, Nov 29 2008]
a(n+1) = a(n) + A110654(n). [From Reinhard Zumkeller, Aug 06 2009]
a(n)=(n*n - 2*n + nmod2)/4 [From Ctibor O. Zizka, Nov 23 2009]
Equals triangle A171608 * ( 1, 2, 3,...) [From Gary W. Adamson, Dec 12 2009]
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
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
a(n) belongs to the sequence iff 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
a(n) is the sum of the positive integers < n that have the opposite parity as n.
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
Third outer diagonal of Losanitsch's Triangle, A034851. [Fred Daniel Kline, Sep 10 2011]
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].
a(n-2) counts the number of distinct graphs with n vertices and n regions. - Erik Hasse, Oct 18 2011
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REFERENCES
| 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.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 25.
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.
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 99.
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.
T. Jenkyns and E. Muller, Triangular triples from ceilings to floors, Amer. Math. Monthly, 107 (Aug. 2000), 634-639.
D. E. Knuth, The art of programming, Vol. 1, 3rd Edition, Addison-Wesley, 1997, Ex. 36 of section 1.2.4.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
J. Nelder, Critical sets in Latin squares, CSIRO Division of Math. and Stats. Newsletter, Vol. 38 (1977), p. 4.
N. Reading, Order Dimension, Strong Bruhat Order and Lattice Properties for Posets, Order, Vol. 19, no. 1 (2002), 73-100.
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]
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Mircea Merca, Inequalities and Identities Involving Sums of Integer Functions, J. Integer Seq., 14(2011), Article 11.9.1.
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LINKS
| Franklin T. Adams-Watters, Table of n, a(n) for n = 0..10000
Washington G. Bomfim, Illustration of the bracelets with 8 beads, 2 of which are red, 1 of which is blue..
H. Bottomley, Illustration of initial terms
P. J. Cameron, BCC Problem List, Problem BCC15.15 (DM285), Discrete Math. 167/168 (1997), 605-615.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 105
V. Jovovic, Vladeta Jovovic, Number of binary matrices
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.
S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, Blending two discrete integrability criteria: ...
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
N. Reading, Order Dimension, Strong Bruhat Order and Lattice Properties for Posets
J. Scholes, 27th Putnam 1966 Prob.A1
N. J. A. Sloane, Classic Sequences
Sam E. Speed, "The Integer Sequence A002620 and Upper Antagonistic Functions" , Journal of Integer Sequences, Vol. 5 (2002), Article 03.1.4
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
Index entries for two-way infinite sequences
Index entries for sequences related to linear recurrences with constant coefficients, signature (2,0,-2,1).
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FORMULA
| a(n)=(2*n^2-1+(-1)^(n))/8. - Paul Barry, May 27 2003
G.f.: x^2/((1-x)^2*(1-x^2)).
E.g.f.: exp(x)*(2*x^2+2*x-1)/8+exp(-x)/8.
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) [From Jaume Oliver Lafont, Dec 05 2008]
a(-n)=a(n).
a(n)=a(n-1)+int(n/2), n>0 - Adam Kertesz (adamkertesz(AT)worldnet.att.net), Sep 20 2000
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
0*0, 0*1, 1*1, 1*2, 2*2, 2*3, 3*3, 3*4, ... with an obvious pattern.
a(n) = sum(floor(k/2), k=1..n) - Yong Kong (ykong(AT)curagen.com), Mar 10 2001
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
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
a(n)=sum{k=0..n, (-1)^(n-k)*C(k, 2) } - Paul Barry, Jul 01 2003
(-1)^n * partial sum of alternating triangular numbers. - Jon Perry, Dec 30 2003
a(n) = A024206(n+1) -n . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 27 2004
Partial sums of A004526. - Lekraj Beedassy, Jun 30 2004
a(n)=a(n-2)+n-1, a(0)=0, a(1)=0. - Paul Barry, Jul 14 2004
a(n+1)=sum min(i, n-i), i=0..n. - Marc LeBrun, Feb 15 2005
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
1 + 1/(1 + 2/(1 + 4/(1 + 6/(1 + 9/(1 + 12/(1 + 16/(1 + . . ))))))) = 6/(Pi^2 - 6) = 1.550546096730... - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 20 2005
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
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
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
a(n) = sum of row (n-2) of triangle A115514. - Gary W. Adamson, Oct 25 2007
For n>1: GreatestCommonDivisor(a(n+1), a(n)) = a(n+1) - a(n). - Reinhard Zumkeller, Apr 06 2008
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
a(n) = SUM((k mod 2)*(n-k): 0<=k<=n), cf. A000035, A001477. [From Reinhard Zumkeller, Nov 05 2009]
a(n) = round((2*n^2-1)/8) = round(n^2/4) = ceil((n^2-1)/4). [From Mircea Merca, Nov 29 2010]
n*a(n+2) = 2*a(n+1)+(n+2)*a(n). Holonomic Ansatz with smallest order of recurrence. [From Thotsaporn Thanatipanonda, Dec 12 2010]
a(n)=(n (2 + n) + n mod 2)/4 [From Fred Daniel Kline, Sep 11, 2011]
a(n) = a199332(n,floor((n+1)/2)). [Reinhard Zumkeller, Nov 23 2011]
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EXAMPLE
| a(3)=2, floor(3/2)*ceiling(3/2)=2
[ n] a(n)
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[ 2] 1
[ 3] 2
[ 4] 1 + 3
[ 5] 2 + 4
[ 6] 1 + 3 + 5
[ 7] 2 + 4 + 6
[ 8] 1 + 3 + 5 + 7
[ 9] 2 + 4 + 6 + 8
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MAPLE
| A002620 := n->floor(n^2/4); G002620 := series(x^2/((1-x)^2*(1-x^2)), x, 60);
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
A002620:=-1/(z+1)/(z-1)^3; [S. Plouffe in his 1992 dissertation, leading zeros dropped.]
A002620 := n -> add(k, k = select(k -> k mod 2 <> n mod 2, [$1 .. n])): seq(A002620(n), n = 0 .. 57);
# Peter Luschny, Jul 06 2011
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MATHEMATICA
| f[n_] := Ceiling[n/2]Floor[n/2]; Table[ f[n], {n, 0, 56}] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 18 2005)
a=0; Table[(a=n^2+n-a)/2, {n, -1, 90}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 18 2009]
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 *)
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PROG
| (MAGMA) [ Floor(n/2)*Ceiling(n/2) : n in [0..40]];
(PARI) a(n)=n^2\4
(PARI) t(n)=n*(n+1)/2 for(i=1, 50, print1(", "(-1)^i*sum(k=1, i, (-1)^k*t(k))))
(PARI) a(n)=n^2>>2 [From Charles R Greathouse IV, Nov 11 2009]
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CROSSREFS
| A087811 is another version of this sequence.
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).
Differences of A002623. Complement of A049068.
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
a(n) = A014616(n-2)+2 = A033638(n)-1 = A078126(n)+1. Cf. A055802, A055803.
Antidiagonal sums of array A003983.
a(2n)=A000290(n) = squares, a(2n+1)=A002378(n) = oblong numbers. A122196 [From Gary W. Adamson, Nov 29 2008]
Cf. A033436, A033437, A033438, A033439, A033440, A033441, A033442, A033443, A033444. [From Reinhard Zumkeller, Nov 30 2009]
Cf. A008233, A008217, A014980, A197081, A197122.
Sequence in context: A088900 A083392 A076921 * A087811 A025699 A194256
Adjacent sequences: A002617 A002618 A002619 * A002621 A002622 A002623
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KEYWORD
| nonn,easy,nice,core
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 11 2009
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