A030179 -id:A030179 - cp's OEIS frontend

A208118 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 0 1 1 vertically.

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 15, 81, 90, 81, 12, 25, 225, 225, 225, 144, 16, 40, 625, 825, 625, 420, 256, 20, 64, 1600, 3025, 3025, 1225, 784, 400, 25, 104, 4096, 9240, 14641, 7315, 2401, 1260, 625, 30, 169, 10816, 28224, 53361, 43681, 17689, 3969, 2025, 900

Offset: 1

R. H. Hardin Feb 23 2012

Table starts
..2...4....6....9....15.....25......40.......64.......104........169........273
..4..16...36...81...225....625....1600.....4096.....10816......28561......74529
..6..36...90..225...825...3025....9240....28224.....93912.....312481.....997815
..9..81..225..625..3025..14641...53361...194481....815409....3418801...13359025
.12.144..420.1225..7315..43681..175560...705600...3503640...17397241...76934095
.16.256..784.2401.17689.130321..577600..2560000..15054400...88529281..443060401
.20.400.1260.3969.34713.303601.1432600..6760000..45648200..308248249.1680152229
.25.625.2025.6561.68121.707281.3553225.17850625.138415225.1073283121.6371392041

			Some solutions for n=4 k=3
..1..1..1....0..1..1....0..0..1....0..0..1....1..1..0....1..0..1....1..1..1
..1..1..1....1..0..1....1..0..1....0..0..1....0..0..1....1..0..1....1..1..1
..0..1..1....0..1..1....0..0..1....0..0..1....1..1..0....1..0..1....1..1..1
..0..1..1....1..0..1....0..0..1....0..0..1....0..0..1....0..0..1....1..0..1

R. H. Hardin, Table of n, a(n) for n = 1..1211

Column 1 is A002620(n+2)
Column 2 is A030179(n+2)
Column 3 is A207363
Row 1 is A006498(n+2)
Row 2 is A189145(n+2)
Row 3 is A207600

A208142 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 1 horizontally and 0 0 1 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 8, 12, 81, 108, 64, 10, 16, 144, 324, 240, 100, 12, 20, 256, 720, 900, 450, 144, 14, 25, 400, 1600, 2400, 2025, 756, 196, 16, 30, 625, 3000, 6400, 6300, 3969, 1176, 256, 18, 36, 900, 5625, 14000, 19600, 14112, 7056, 1728, 324, 20, 42, 1296

Offset: 1

R. H. Hardin Feb 23 2012

Table starts
..2...4....6.....9....12.....16.....20......25......30.......36.......42
..4..16...36....81...144....256....400.....625.....900.....1296.....1764
..6..36..108...324...720...1600...3000....5625....9450....15876....24696
..8..64..240...900..2400...6400..14000...30625...58800...112896...197568
.10.100..450..2025..6300..19600..49000..122500..264600...571536..1111320
.12.144..756..3969.14112..50176.141120..396900..952560..2286144..4889808
.14.196.1176..7056.28224.112896.352800.1102500.2910600..7683984.17929296
.16.256.1728.11664.51840.230400.792000.2722500.7840800.22581504.57081024

			Some solutions for n=4 k=3
..1..0..0....0..0..0....0..1..0....1..1..1....0..1..0....0..1..0....0..1..0
..0..0..0....1..0..0....1..1..0....1..1..0....1..0..0....0..1..0....1..0..1
..0..0..0....0..0..0....1..1..0....1..0..0....1..0..0....0..1..0....0..0..0
..0..0..0....0..0..0....1..0..0....1..0..0....1..0..0....0..1..0....0..0..0

R. H. Hardin, Table of n, a(n) for n = 1..4508

Column 1 is A004275(n+1)
Column 2 is A016742
Column 3 is A202195(n-2)
Row 1 is A002620(n+2)
Row 2 is A030179(n+2)
Row 3 is A202093(n-2)

Empirical for column k:
k=1: a(n) = 2*n
k=2: a(n) = 4*n^2
k=3: a(n) = 3*n^3 + 3*n^2
k=4: a(n) = (9/4)*n^4 + (9/2)*n^3 + (9/4)*n^2
k=5: a(n) = n^5 + 4*n^4 + 5*n^3 + 2*n^2
k=6: a(n) = (4/9)*n^6 + (8/3)*n^5 + (52/9)*n^4 + (16/3)*n^3 + (16/9)*n^2
k=7: a(n) = (5/36)*n^7 + (5/4)*n^6 + (155/36)*n^5 + (85/12)*n^4 + (50/9)*n^3 + (5/3)*n^2

A208555 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 10, 36, 36, 9, 16, 100, 90, 81, 12, 26, 256, 330, 225, 144, 16, 42, 676, 1008, 1089, 420, 256, 20, 68, 1764, 3354, 3969, 2508, 784, 400, 25, 110, 4624, 10710, 16641, 10080, 5776, 1260, 625, 30, 178, 12100, 34884, 65025, 50052, 25600, 11020

Offset: 1

R. H. Hardin Feb 28 2012

Table starts
..2...4....6....10.....16.....26......42.......68.......110........178
..4..16...36...100....256....676....1764.....4624.....12100......31684
..6..36...90...330...1008...3354...10710....34884....112530.....364722
..9..81..225..1089...3969..16641...65025...263169...1046529....4198401
.12.144..420..2508..10080..50052..221340..1042416...4742628...21989868
.16.256..784..5776..25600.150544..753424..4129024..21492496..115175824
.20.400.1260.11020..52000.351140.1913940.11836400..67894220..407225740
.25.625.2025.21025.105625.819025.4862025.33930625.214476025.1439823025

			Some solutions for n=4 k=3
..1..1..0....1..1..0....1..1..0....1..1..1....1..0..1....1..0..1....0..1..1
..1..0..0....0..1..0....0..1..0....1..1..0....0..1..0....1..0..1....0..1..1
..1..0..0....1..0..0....0..1..0....1..0..1....1..0..0....1..0..0....0..1..0
..1..0..0....0..1..0....0..1..0....1..1..0....0..1..0....1..0..0....0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..2132

Column 1 is A002620(n+2)
Column 2 is A030179(n+2)
Column 3 is A207363
Row 1 is A006355(n+2)
Row 2 is A206981
Row 3 is A207454

A207949 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 1 horizontally and 0 1 0 and 1 0 1 vertically.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 10, 12, 81, 102, 100, 16, 16, 144, 289, 370, 256, 26, 20, 256, 612, 1369, 1232, 676, 42, 25, 400, 1296, 3478, 5929, 4238, 1764, 68, 30, 625, 2340, 8836, 18172, 26569, 14406, 4624, 110, 36, 900, 4225, 18330, 55696, 98126, 117649

Offset: 1

R. H. Hardin Feb 21 2012

Table starts
..2....4.....6......9......12.......16.......20........25........30.........36
..4...16....36.....81.....144......256......400.......625.......900.......1296
..6...36...102....289.....612.....1296.....2340......4225......6890......11236
.10..100...370...1369....3478.....8836....18330.....38025.....69420.....126736
.16..256..1232...5929...18172....55696...133812....321489....662256....1364224
.26..676..4238..26569...98126...362404..1007146...2798929...6501278...15100996
.42.1764.14406.117649..524104..2334784..7513176..24176889..63380130..166152100
.68.4624.49164.522729.2806686.15069924.56114310.208947025.617864520.1827049536

			Some solutions for n=4 k=3
..1..1..0....1..1..0....0..0..0....1..0..0....0..1..0....0..1..0....0..1..0
..0..0..0....1..0..1....1..0..0....0..0..0....1..0..0....1..0..1....0..1..0
..0..0..0....1..0..1....1..1..1....0..0..0....1..0..1....1..0..1....0..1..0
..1..0..0....0..1..0....1..1..1....0..1..0....1..1..1....1..1..0....0..1..0

R. H. Hardin, Table of n, a(n) for n = 1..391

Column 1 is A006355(n+2)
Column 2 is A206981
Column 3 is A207249
Column 4 is A207854
Row 1 is A002620(n+2)
Row 2 is A030179(n+2)
Row 3 is A207118

A014540 Rectilinear crossing number of complete graph on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 9, 19, 36, 62, 102, 153, 229, 324, 447, 603, 798, 1029, 1318, 1657, 2055, 2528, 3077, 3699, 4430, 5250, 6180

Offset: 1

Eric W. Weisstein

The values a(19) and a(21) were obtained by Aichholzer et al. in 2006. The value a(18) is claimed by the Rectilinear Crossing Number project after months of distributed computing. This was confirmed by Abrego et al., they also found the values a(20) and a(22) to a(27). The next unknown entry, a(28), is either 7233 or 7234. - Bernardo M. Abrego (bernardo.abrego(AT)csun.edu), May 05 2008

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.18, p. 532.
M. Gardner, Crossing Numbers. Ch. 11 in Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, 1986.
C. Thomassen, Embeddings and minors, pp. 301-349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.

B. M. Abrego, S. Fernandez-Merchant, J. Leaños, and G. Salazar, The maximum number of halving lines and the rectilinear crossing number of K_n for n <= 27, Electronic Notes in Discrete Mathematics, 30 (2008), 261-266.
O. Aichholzer, Crossing number project.
O. Aichholzer, F. Aurenhammer, and H. Krasser, Progress on rectilinear crossing numbers. [Broken link]
O. Aichholzer, F. Aurenhammer, and H. Krasser, Progress on rectilinear crossing numbers, Technical report, IGI-TU Graz, Austria, 2001.
O. Aichholzer, F. Aurenhammer, and H. Krasser, On the Rectilinear Crossing Number. [Broken link]
O. Aichholzer, J. Garcia, D. Orden, and P. Ramos, New lower bounds for the number of <= k-edges and the rectilinear crossing number of K_n, Discrete & Computational Geometry 38 (2007), 1-14.
O. Aichholzer and H. Krasser, The point set order type data base: a collection of applications and results, pp. 17-20 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001. [Broken link]
D. Archdeacon, The rectilinear crossing number.
D. Bienstock and N. Dean, Bounds for rectilinear crossing numbers, J. Graph Theory 17 (1993) 333-348
A. Brodsky, S. Durocher, and E. Gethner, The Rectilinear Crossing Number of K_{10} is 62, The Electronic J. Combin, #R23, 2001.
A. Brodsky, S. Durocher, and E. Gethner, Toward the rectilinear crossing number of K_n: new drawings, upper bounds, and asymptotics, Discrete Math. 262 (2003), 59-77.
D. Garber, The Orchard crossing number of an abstract graph, arXiv:math/0303317 [math.CO], 2003-2009.
H. F. Jensen, An Upper Bound for the Rectilinear Crossing Number of the Complete Graph, J. Comb. Th. Ser. B 10, 212-216, 1971.
Eric Weisstein's World of Mathematics, Graph Crossing Number.
Eric Weisstein's World of Mathematics, Rectilinear Crossing Number.
Eric Weisstein's World of Mathematics, Zarankiewicz's Conjecture.

Cf. A000241, A030179, A006247.

102 from Oswin Aichholzer (oswin.aichholzer(AT)tugraz.at), Aug 14 2001
153 from Hannes Krasser (hkrasser(AT)igi.tu-graz.ac.at), Sep 17 2001
More terms from Eric W. Weisstein, Nov 30 2006
More terms from Bernardo M. Abrego (bernardo.abrego(AT)csun.edu), May 05 2008

A250432 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing sum of every two consecutive values in every row and column.

Original entry on oeis.org

16, 36, 36, 81, 108, 81, 144, 324, 324, 144, 256, 720, 1296, 720, 256, 400, 1600, 3600, 3600, 1600, 400, 625, 3000, 10000, 12000, 10000, 3000, 625, 900, 5625, 22500, 40000, 40000, 22500, 5625, 900, 1296, 9450, 50625, 105000, 160000, 105000, 50625, 9450

Offset: 1

R. H. Hardin, Nov 22 2014

Table starts
...16....36.....81.....144......256.......400.......625........900........1296
...36...108....324.....720.....1600......3000......5625.......9450.......15876
...81...324...1296....3600....10000.....22500.....50625......99225......194481
..144...720...3600...12000....40000....105000....275625.....617400.....1382976
..256..1600..10000...40000...160000....490000...1500625....3841600.....9834496
..400..3000..22500..105000...490000...1715000...6002500...17287200....49787136
..625..5625..50625..275625..1500625...6002500..24010000...77792400...252047376
..900..9450..99225..617400..3841600..17287200..77792400..280052640..1008189504
.1296.15876.194481.1382976..9834496..49787136.252047376.1008189504..4032758016
.1764.24696.345744.2765952.22127616.124467840.700131600.3080579040.13554547776
Essentially the same as A202100; the mapping between the binary arrays in both sequences is by flipping all entries in one set of arrays. - Joerg Arndt, Dec 01 2014

			Some solutions for n=5 k=4
..0..0..0..0..1....0..0..0..0..0....0..0..1..0..1....0..0..0..1..1
..0..1..0..1..1....0..0..0..1..0....1..1..1..1..1....0..0..1..1..1
..0..0..1..0..1....0..0..0..1..0....0..0..1..1..1....0..0..1..1..1
..0..1..1..1..1....0..0..0..1..0....1..1..1..1..1....0..0..1..1..1
..0..1..1..1..1....0..0..0..1..0....0..1..1..1..1....1..0..1..1..1
..0..1..1..1..1....0..0..0..1..1....1..1..1..1..1....1..1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..1104

Column 1 is A030179(n+3), A202093 - A202099 (further columns).

Empirical for column k:
k=1: a(n) = 2*a(n-1) +2*a(n-2) -6*a(n-3) +6*a(n-5) -2*a(n-6) -2*a(n-7) +a(n-8); also a polynomial of degree 4 plus a quasipolynomial of degree 2 with period 2
k=2: [order 12; also a polynomial of degree 6 plus a quasipolynomial of degree 4 with period 2]
k=3: [order 16; also a polynomial of degree 8 plus a quasipolynomial of degree 6 with period 2]
k=4: [order 20; also a polynomial of degree 10 plus a quasipolynomial of degree 8 with period 2]
k=5: [order 24; also a polynomial of degree 12 plus a quasipolynomial of degree 10 with period 2]
k=6: [order 28; also a polynomial of degree 14 plus a quasipolynomial of degree 12 with period 2]
k=7: [order 32; also a polynomial of degree 16 plus a quasipolynomial of degree 14 with period 2]

A267719 T(n,k)=Number of nXk 0..1 arrays with every repeated value in every row and column greater than the previous repeated value.

Original entry on oeis.org

2, 4, 4, 6, 16, 6, 9, 36, 36, 9, 12, 81, 102, 81, 12, 16, 144, 270, 270, 144, 16, 20, 256, 546, 872, 546, 256, 20, 25, 400, 1080, 1915, 1915, 1080, 400, 25, 30, 625, 1866, 4266, 4444, 4266, 1866, 625, 30, 36, 900, 3186, 7879, 10489, 10489, 7879, 3186, 900, 36, 42, 1296

Offset: 1

R. H. Hardin, Jan 19 2016

Table starts
..2....4....6.....9.....12.....16.....20......25......30.......36.......42
..4...16...36....81....144....256....400.....625.....900.....1296.....1764
..6...36..102...270....546...1080...1866....3186....5010.....7830....11550
..9...81..270...872...1915...4266...7879...14632...24440....40816....63714
.12..144..546..1915...4444..10489..20226...39485...68430...119560...192852
.16..256.1080..4266..10489..26906..54887..114392..209470...385962...656670
.20..400.1866..7879..20226..54887.116036..255009..481390...932245..1627470
.25..625.3186.14632..39485.114392.255009..592110.1179367..2391952..4403235
.30..900.5010.24440..68430.209470.481390.1179367.2411426..5148115..9689432
.36.1296.7830.40816.119560.385962.932245.2391952.5148115.11438402.22656990

			Some solutions for n=5 k=4
..1..0..0..1....0..1..1..0....1..1..0..1....0..1..1..0....1..0..0..1
..1..0..1..1....1..0..0..1....0..1..1..0....1..0..0..1....1..0..1..1
..0..1..0..0....0..1..1..0....1..0..1..1....0..0..1..0....0..1..0..0
..1..1..0..1....1..0..1..1....0..1..0..0....0..1..0..0....1..0..1..1
..0..0..1..0....1..1..0..1....1..0..1..0....1..1..0..1....0..1..0..0

R. H. Hardin, Table of n, a(n) for n = 1..543

Column 1 is A002620(n+2).

Column 2 is A030179(n+2).

Empirical for column k:
k=1: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4)
k=2: a(n) = 2*a(n-1) +2*a(n-2) -6*a(n-3) +6*a(n-5) -2*a(n-6) -2*a(n-7) +a(n-8)
k=3: [order 12]
k=4: [order 16] for n>18
k=5: [order 20] for n>22
k=6: [order 24] for n>27
k=7: [order 28] for n>30

A212892 a(n) = n^4/8 if n is even, a(n) = (n^2-1)^2/8 if n is odd.

Original entry on oeis.org

0, 0, 2, 8, 32, 72, 162, 288, 512, 800, 1250, 1800, 2592, 3528, 4802, 6272, 8192, 10368, 13122, 16200, 20000, 24200, 29282, 34848, 41472, 48672, 57122, 66248, 76832, 88200, 101250, 115200, 131072, 147968, 167042, 187272, 209952, 233928, 260642, 288800, 320000, 352800, 388962, 426888

Offset: 0

Clark Kimberling, May 30 2012, and Bill Correll, Jr., Jun 07 2014

The sequence a(0)=0, a(1)=2, a(2)=8, a(3)=32, ... arises as the number of quadruples (w,x,y,z) with all terms in {0,...,n} and w-x, x-y, and y-z all odd. For a guide to related sequences, see A211795.
The sequence a(3)=2, a(4)=8, a(5)=32, ... is the number of L3 configurations in an n X n permutation array. An L3 configuration is defined to be a set of 3 equally-spaced, collinear points in a permutation array.  L3 configurations were first enumerated by Davies in his study of the density of Costas arrays. They constitute a violation of the definition of a Costas array, so Costas arrays cannot have any. Davies's work went unpublished until it appeared in the survey paper by Drakakis. - Bill Correll, Jr., Jun 07 2014
Every term is even.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
K. Drakakis, A review of Costas arrays, Journal of Applied Mathematics, pp. 1-32, 2006, Article ID 26385.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A211795.

Cf. A030179.

f:=n->if n mod 2 = 0 then n^4/8 else (n^2-1)^2/8; fi; [seq(f(n),n=0..50)]; # N. J. A. Sloane, Mar 06 2015

(* If the sequence begins a(0)=0, a(1)=2, a(2)=8, a(3)=32, ... *)
t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[Mod[w - x, 2] == Mod[x - y, 2] == Mod[y - z, 2] == 1, s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 40]]   (* A212892 *)
m/2 (* integers *)

If the sequence begins a(0)=0, a(1)=2, a(2)=8, a(3)=32, ...: a(n) = 2*a(n-1)+2*a(n-2)-6*a(n-3)+6*a(n-5)-2*a(n-6)-2*a(n-7)+a(n-8). G.f.: f(x)/g(x), where f(x) = x+2*x^2+6*x^3+2*x^4+x^5 and g(x) = ((1-x)^5)*(1+x)^3.
Sum_{n>=2} 1/a(n) = Pi^4/180 + Pi^2/6 - 3/2. - Amiram Eldar, Sep 08 2022
a(n) = 2*floor(n^2/4)^2 = 2*A030179(n). - Ridouane Oudra, Sep 12 2023

Formed by merging two entries that arose in different contexts. Thanks to Alois P. Heinz, Mar 04 2015 for noticing that the sequences were essentially identical. - N. J. A. Sloane, Mar 06 2015

A298368 Triangle read by rows: T(n, k) = floor((n-1)/2)floor(n/2)floor((k-1)/2)*floor(k/2).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 2, 4, 0, 0, 4, 8, 16, 0, 0, 6, 12, 24, 36, 0, 0, 9, 18, 36, 54, 81, 0, 0, 12, 24, 48, 72, 108, 144, 0, 0, 16, 32, 64, 96, 144, 192, 256, 0, 0, 20, 40, 80, 120, 180, 240, 320, 400, 0, 0, 25, 50, 100, 150, 225, 300, 400, 500, 625

Offset: 1

Eric W. Weisstein, Jan 17 2018

T(n, k) is conjectured by Zarankiewicz's conjecture to be the crossing number of the complete bipartite graph K_{k,n}.

			First rows are given by:
  0;
  0,   0;
  0,   0,   1;
  0,   0,   2,   4;
  0,   0,   4,   8,  16;
  0,   0,   6,  12,  24,  36;
  0,   0,   9,  18,  36,  54,  81;
  0,   0,  12,  24,  48,  72, 108, 144;
  0,   0,  16,  32,  64,  96, 144, 192, 256;
  0,   0,  20,  40,  80, 120, 180, 240, 320, 400;

Robert Israel, Table of n, a(n) for n = 1..10011 (rows 1 to 141, flattened)
Eric Weisstein's World of Mathematics, Graph Crossing Number.
Eric Weisstein's World of Mathematics, Zarankiewicz's Conjecture.

Cf. A030179, A191928.

seq(seq(floor((k-1)/2)*floor(k/2)*floor((n-1)/2)*floor(n/2),k=1..n),n=1..12); # Robert Israel, Jan 17 2018

Table[Floor[(m - 1)/2] Floor[m/2] Floor[(n - 1)/2] Floor[n/2], {n, 11}, {m, n}] // Flatten
Table[Times @@ Floor[{m, m - 1, n, n - 1}/2], {n, 11}, {m, n}] // Flatten

T(n,n) = A030179(n).
From Robert Israel, Jan 17 2018: (Start)
T(n,k) = A002620(n-1)*A002620(k-1).
G.f. as triangle: x^3*y^3*(1+2*x*y+6*x^2*y^2-4*x^3*y-8*x^3*y^2+2*x^4*y+2*x^3*y^3-4*x^4*y^2-2*x^4*y^3+4*x^5*y^2+ x^4*y^4-4*x^5*y^3-2*x^5*y^4+4*x^6*y^3+2*x^7*y^4)/
  ((1-x*y)^5*(1+x*y)^3*(1-x)^3*(1+x)). (End)

cp's OEIS Frontend

A208118 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 0 1 and 0 1 1 vertically.

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A208142 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 1 horizontally and 0 0 1 and 1 0 1 vertically.

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A208555 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 0 and 0 0 1 horizontally and 0 0 1 and 0 1 1 vertically.

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A207949 T(n,k)=Number of nXk 0..1 arrays avoiding 0 0 1 and 0 1 1 horizontally and 0 1 0 and 1 0 1 vertically.

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A014540 Rectilinear crossing number of complete graph on n nodes.

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A250432 T(n,k)=Number of (n+1)X(k+1) 0..1 arrays with nondecreasing sum of every two consecutive values in every row and column.

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A267719 T(n,k)=Number of nXk 0..1 arrays with every repeated value in every row and column greater than the previous repeated value.

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A212892 a(n) = n^4/8 if n is even, a(n) = (n^2-1)^2/8 if n is odd.

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A298368 Triangle read by rows: T(n, k) = floor((n-1)/2)*floor(n/2)*floor((k-1)/2)*floor(k/2).

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A298368 Triangle read by rows: T(n, k) = floor((n-1)/2)floor(n/2)floor((k-1)/2)*floor(k/2).