%I A002620 M0998 N0374
%S A002620 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 A002620 144,156,169,182,196,210,225,240,256,272,289,306,324,342,361,380,400,
%U A002620 420,441,462,484,506,529,552,576,600,625,650,676,702,729,756,784,812
%N A002620 Quarter-squares: floor(n/2)*ceiling(n/2). Equivalently, floor(n^2/4).
%C A002620 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 2-covers
of an n-set; 2 X n binary matrices with no zero columns up to row
and column permutation - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun
08, 2000.
%C A002620 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 A002620 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 A002620 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 A002620 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
%C A002620 Sums of pairs of neighboring terms are triangular numbers in increasing
order. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 19 2002
%C A002620 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 (rshepherd2(AT)hotmail.com), Sep 17 2002
%C A002620 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 (rshepherd2(AT)hotmail.com), Sep
17 2002
%C A002620 Terms are the geometric mean and arithmetic mean of their neighbors alternately.
- Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 17 2002
%C A002620 Maximum product of two integers whose sum is n. - Matthew Vandermast
(ghodges14(AT)comcast.net), Mar 04 2003
%C A002620 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 (perry(AT)globalnet.co.uk),
Jul 08 2003
%C A002620 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 (perry(AT)globalnet.co.uk),
Jul 08 2003
%C A002620 Union of square numbers (A000290) and oblong numbers (A002378). - Lekraj
Beedassy (blekraj(AT)yahoo.com), Oct 02 2003
%C A002620 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 A002620 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
%C A002620 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 (deutsch(AT)duke.poly.edu), Jan 12 2004
%C A002620 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 (rshepherd2(AT)hotmail.com),
Feb 27 2004
%C A002620 See A092186 for another application.
%C A002620 Also, the number of nonisomorphic transversal combinatorial geometries
of rank 2. - Alexandr S. Radionov (rasmailru(AT)mail.ru), Jun 02
2004
%C A002620 a(n+1) is the transform of n under the Riordan array (1/(1-x^2),x). -
Paul Barry (pbarry(AT)wit.ie), Apr 16 2005
%C A002620 a(n) = A108561(n+1,n-2) for n>2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Jun 10 2005
%C A002620 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
%C A002620 a(n) = Sum(Min{k,n-k}: 0<=k<=n), sums of rows of the triangle in A004197.
- Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 27 2005
%C A002620 a(n+1) is the number of noncongruent integer-sided triangles with largest
side n - David W. Wilson. [Comment corrected Sep 26 2006]
%C A002620 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 A002620 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 A002620 Maximal number of squares (maximal area) in a polyomino with perimeter
2n. - Tanya Khovanova (tanyakh(AT)yahoo.com), Jul 04 2007
%C A002620 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
%C A002620 Equals row sums of triangle A122196 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Nov 29 2008]
%C A002620 a(n+1) = a(n) + A110654(n). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Aug 06 2009]
%C A002620 a(n)=(n*n - 2*n + nmod2)/4 [From Ctibor O. Zizka (c.zizka(AT)email.cz),
Nov 23 2009]
%C A002620 Equals triangle A171608 * ( 1, 2, 3,...) [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Dec 12 2009]
%D A002620 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 A002620 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag,
1976, page 73, problem 25.
%D A002620 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 A002620 E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon
test, Annals Math. Stat., 26 (1955), 301-312.
%D A002620 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley,
Reading, MA, 1990, p. 99.
%D A002620 T. Jenkyns and E. Muller, Triangular triples from ceilings to floors,
Amer. Math. Monthly, 107 (Aug. 2000), 634-639.
%D A002620 D. E. Knuth, The art of programming, Vol. 1, 3rd Edition, Addison-Wesley,
1997, Ex. 36 of section 1.2.4.
%D A002620 S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe,
Chem. Ber. 30 (1897), 1917-1926.
%D A002620 J. Nelder, Critical sets in Latin squares, CSIRO Division of Math. and
Stats. Newsletter, Vol. 38 (1977), p. 4.
%D A002620 N. Reading, Order Dimension, Strong Bruhat Order and Lattice Properties
for Posets, Order, Vol. 19, no. 1 (2002), 73-100.
%D A002620 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 A002620 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A002620 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%H A002620 Franklin T. Adams-Watters, <a href="b002620.txt">Table of n, a(n) for
n = 0..10000</a>
%H A002620 H. Bottomley, <a href="a2620.gif">Illustration of initial terms</a>
%H A002620 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 A002620 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 A002620 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=105">
Encyclopedia of Combinatorial Structures 105</a>
%H A002620 V. Jovovic, Vladeta Jovovic, <a href="a005748.PDF">Number of binary matrices</
a>
%H A002620 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 A002620 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 A002620 S. 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 A002620 S. 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 A002620 N. Reading, <a href="http://www.math.umn.edu/~reading/dissective.ps">
Order Dimension, Strong Bruhat Order and Lattice Properties for Posets
</a>
%H A002620 J. Scholes, <a href="http://www.kalva.demon.co.uk/putnam/psoln/psol661.html">
27th Putnam 1966 Prob.A1</a>
%H A002620 N. J. A. Sloane, <a href="classic.html#LOSS">Classic Sequences</a>
%H A002620 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 A002620 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 A002620 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 A002620 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
a>
%H A002620 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%F A002620 a(n)=a(n-1)+int(n/2), n>0 - Adam Kertesz (adamkertesz(AT)worldnet.att.net),
Sep 20 2000
%F A002620 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 (se16(AT)btinternet.com), Mar 08 2000
%F A002620 0*0, 0*1, 1*1, 1*2, 2*2, 2*3, 3*3, 3*4, ... with an obvious pattern.
%F A002620 a(n) = sum(floor(k/2), k=1..n) - Yong Kong (ykong(AT)curagen.com), Mar
10 2001
%F A002620 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 A002620 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 A002620 a(n)=(2n^2-1+(-1)^(n))/8. - Paul Barry (pbarry(AT)wit.ie), May 27 2003
%F A002620 a(n)=sum{k=0..n, (-1)^(n-k)C(k, 2) } - Paul Barry (pbarry(AT)wit.ie),
Jul 01 2003
%F A002620 G.f.: x^2/((1-x)^2(1-x^2)). E.g.f.: exp(x)(2x^2+2x-1)/8+exp(-x)/8. a(-n)=a(n).
%F A002620 (-1)^n * partial sum of alternating triangular numbers. - Jon Perry (perry(AT)globalnet.co.uk),
Dec 30 2003
%F A002620 a(n) = A024206(n+1) -n . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr),
Feb 27 2004
%F A002620 Partial sums of A004526. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jun
30 2004
%F A002620 a(n)=a(n-2)+n-1, a(0)=0, a(1)=0. - Paul Barry (pbarry(AT)wit.ie), Jul
14 2004
%F A002620 a(n+1)=sum min(i, n-i), i=0..n. - Marc LeBrun (mlb(AT)well.com), Feb
15 2005
%F A002620 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 (pbarry(AT)wit.ie), Apr 16 2005
%F A002620 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
%F A002620 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 (jvospost3(AT)gmail.com), Jan 19 2006
%F A002620 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 (qntmpkt(AT)yahoo.com), Jul 27 2007
%F A002620 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 (thomas.wieder(AT)t-online.de),
Sep 01 2007
%F A002620 a(n) = sum of row (n-2) of triangle A115514. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Oct 25 2007
%F A002620 For n>1: GreatestCommonDivisor(a(n+1), a(n)) = a(n+1) - a(n). - Reinhard
Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 06 2008
%F A002620 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 A002620 Linear recurrence: a(n)=2a(n-1)-2a(n-3)+a(n-4) [From Jaume Oliver Lafont
(joliverlafont(AT)gmail.com), Dec 05 2008]
%F A002620 a(n) = SUM((k mod 2)*(n-k): 0<=k<=n), cf. A000035, A001477. [From Reinhard
Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 05 2009]
%e A002620 a(3)=2, floor(3/2)*ceiling(3/2)=2
%p A002620 A002620 := n->floor(n^2/4); G002620 := series(x^2/((1-x)^2*(1-x^2)),x,
60);
%p A002620 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 A002620:=-1/(z+1)/(z-1)^3; [S. Plouffe in his 1992 dissertation, leading
zeros dropped.]
%t A002620 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)
%t A002620 a=0;Table[(a=n^2+n-a)/2,{n,-1,90}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com),
Nov 18 2009]
%o A002620 (MAGMA) [ Floor(n/2)*Ceiling(n/2) : n in [0..40]];
%o A002620 (PARI) a(n)=n^2\4
%o A002620 (PARI) t(n)=n*(n+1)/2 for(i=1,50,print1(","(-1)^i*sum(k=1,i,(-1)^k*t(k))))
%o A002620 (PARI) a(n)=n^2>>2 [From Charles R Greathouse IV (charles.greathouse(AT)case.edu),
Nov 11 2009]
%Y A002620 A087811 is another version of this sequence.
%Y A002620 First differences give integers repeated (cf. A008619 or A004526).
%Y A002620 Differences of A002623. Complement of A049068. Cf. A005044, A030179.
%Y A002620 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
%Y A002620 Cf. A024206, A072280, A002984, A007590, A000212, A118015, A056827, A118013.
%Y A002620 a(n) = A014616(n-2)+2 = A033638(n)-1 = A078126(n)+1. Cf. A055802, A055803.
%Y A002620 Antidiagonal sums of array A003983.
%Y A002620 a(2n)=A000290(n) = squares, a(2n+1)=A002378(n) = oblong numbers.
%Y A002620 Cf. A128174.
%Y A002620 Cf. A000601.
%Y A002620 Cf. A115514.
%Y A002620 A122196 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 29 2008]
%Y A002620 Cf. A033436, A033437, A033438, A033439, A033440, A033441, A033442, A033443,
A033444. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Nov 30 2009]
%Y A002620 Cf. A171608 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 12 2009]
%Y A002620 Sequence in context: A088900 A083392 A076921 this_sequence A087811 A025699
A024617
%Y A002620 Adjacent sequences: A002617 A002618 A002619 this_sequence A002621 A002622
A002623
%K A002620 nonn,easy,nice,core,new
%O A002620 0,4
%A A002620 N. J. A. Sloane (njas(AT)research.att.com).
%E A002620 Removed attribute "conjectured" from Plouffe g.f R. J. Mathar (mathar(AT)strw.leidenuniv.nl),
Mar 11 2009
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