A000241 -id:A000241 - cp's OEIS frontend

A028723 a(n) = (1/4)floor(n/2)floor((n-1)/2)floor((n-2)/2)floor((n-3)/2).

0, 0, 0, 0, 0, 1, 3, 9, 18, 36, 60, 100, 150, 225, 315, 441, 588, 784, 1008, 1296, 1620, 2025, 2475, 3025, 3630, 4356, 5148, 6084, 7098, 8281, 9555, 11025, 12600, 14400, 16320, 18496, 20808, 23409, 26163, 29241, 32490, 36100, 39900, 44100, 48510, 53361, 58443

Offset: 0

N. J. A. Sloane

It is not known whether A000241 and this sequence agree.
Conjectured to be crossing number of complete graph K_n, see A000241.
a(n+1) is the maximum number of rectangles that can be formed from n lines. - Erich Friedman
Number of symmetric Dyck paths of semilength n and having five peaks. E.g., a(6)=3 because we have U*DU*DUU*DDU*DU*D, U*DUU*DU*DU*DDU*D and UU*DU*DU*DU*DU*DD, where U=(1,1), D=(1,-1) and * indicates a peak. - Emeric Deutsch, Jan 12 2004
a(n-5) is the number of length n words, w(1), w(2), ..., w(n) on alphabet {0,1,2} such that w(i) >= w(i+2) for all i. - Geoffrey Critzer, Mar 15 2014
a(n-1) is the number of length n binary strings beginning with a 1 that have exactly two pairs of consecutive 0's and two pairs of consecutive 1's. - Jeremy Dover, Jul 04 2016
Consider the partitions of n into two parts (p,q). Then 2*a(n+2) represents the total volume of all rectangular prisms with dimensions p, q and |q - p|. - Wesley Ivan Hurt, Apr 12 2018
a(n+1) is the number of subsets of {1, 2, ..., n} that contain 2 odd and 2 even numbers.  For example, for n = 6, a(7) = 9 and the 9 subsets are {1,2,3,4}, {1,2,3,6}, {1,2,4,5}, {1,2,5,6}, {1,3,4,6}, {1,4,5,6}, {2,3,4,5}, {2,3,5,6}, {3,4,5,6}. - Enrique Navarrete, Dec 22 2019
a(n+1) is the maximum number of induced 4-cycles in an n-node graph (Pippenger and Golumbic 1975). - Pontus von Brömssen, Mar 27 2022

			G.f. = x^5 + 3*x^6 + 9*x^7 + 18*x^8 + 36*x^9 + 60*x^10 + 100*x^11 + ...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.18, p. 533.
Martin Gardner, Knotted Doughnuts and Other Mathematical Entertainments, W. H. Freeman & Company, 1986, Chapter 11, pages 133-144.
Carsten Thomassen, Embeddings and Minors, in: R. L. Graham, M. Grötschel, and L. Lovász, Handbook of Combinatorics, Vol. 1, Elsevier, 1995, p. 314.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Bernardo M. Abrego, Oswin Aichholzer, Silvia Fernandez-Merchant, Pedro Ramos, and Gelasio Salazar, The 2-Page Crossing Number of K_n, arXiv:1206.5669 [math.CO], 2012.
Bernardo M. Abrego, Oswin Aichholzer, Silvia Fernandez-Merchant, Pedro Ramos, and Gelasio Salazar, The 2-Page Crossing Number of K_n, Discrete Comput. Geom., Vol. 49, No. 4 (2013), pp. 747-777. MR3068573.
James Dolan et al., su(3) and Zarankiewicz's conjecture.
Dhruv Mubayi, Counting substructures II: Hypergraphs, Combinatorica, Vol. 33, No. 5 (2013), pp. 591--612. MR3132928
Nicholas Pippenger and Martin Charles Golumbic, The inducibility of graphs, Journal of Combinatorial Theory Series B 19 (1975), 189-203.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A000241, A006918.

[(n^4-8*n^3+18*n^2-12*n+2*n*(n-2)*((1+(-1)^n)/2)+(2*n-3)^2*((1-(-1)^n)/2))/64: n in [0..50]]; // Vincenzo Librandi, Mar 23 2014

A028723:=n->(1/4)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2)*floor((n-3)/2); seq(A028723(n), n=0..100); # Wesley Ivan Hurt, Nov 01 2013

Table[If[EvenQ[n], n(n-2)^2(n-4)/64, (n-1)^2(n-3)^2/64], {n, 0, 50}]
Table[(n^4 -8n^3 +18n^2 -12n + 2n(n-2)((1+(-1)^n)/2) +(2n-3)^2((1-(-1)^n)/2))/64, {n, 0, 50}] (* Vincenzo Librandi, Mar 23 2014 *)
LinearRecurrence[{2, 2,-6,0,6,-2,-2,1}, {0,0,0,0,0,1,3,9}, 50] (* Harvey P. Dale, Sep 13 2018 *)
Times@@@Table[Floor[(n-k)/2], {n,0,60}, {k,0,3}]/4 (* Eric W. Weisstein, Apr 29 2019 *)

a(n) = if (n % 2, (n-1)^2 *(n-3)^2/64, n*(n-2)^2 *(n-4)/64); \\ Michel Marcus, Nov 02 2013

{a(n) = prod(k=0, 3, (n - k) \ 2) / 4}; /* Michael Somos, Nov 02 2014 */

[(n*(-12 +18*n -8*n^2 +n^3) +2*n*(n-2)*((n+1)%2) +(2*n-3)^2*(n%2))/64 for n in (0..60)] # G. C. Greubel, Apr 08 2022

If n even, n*(n-2)^2*(n-4)/64; if n odd, (n-1)^2*(n-3)^2/64.
G.f.: x^5*(1+x+x^2)/((1-x)^5*(1+x)^3). - Emeric Deutsch, Jan 12 2004
For n>2, a(n) = A007590(n-3)*A007590(n-1)/16. - Richard R. Forberg, Dec 03 2013
a(n) = (n^4 -8*n^3 +18*n^2 -12*n +2*n*(n-2)*((1+(-1)^n)/2) + (2*n-3)^2*((1-(-1)^n)/2))/64. - Luce ETIENNE, Mar 22 2014
Euler transform of length 3 sequence [3, 3, -1]. - Michael Somos, Nov 02 2014
a(n) = a(4-n) for all n in Z. - Michael Somos, Nov 02 2014
0 = -3 + a(n) - a(n+1) - 3*a(n+2) + 3*a(n+3) + 3*a(n+4) - 3*a(n+5) - a(n+6) + a(n+7) for all n in Z. - Michael Somos, Nov 02 2014
0 = a(n)*(+a(n+2) + a(n+3)) + a(n+1)*(-3*a(n+2) +a(n+3)) for all n in Z. - Michael Somos, Nov 02 2014
a(n+1)^2 - a(n)*a(n+2) = binomial(n/2, 2)^3 for all even n in Z ( = 0 if n odd). - Michael Somos, Nov 02 2014
a(n)*(a(n+1) + a(n+2)) +a(n+1)*(-3*a(n+1) + a(n+2)) = 0 for all even n in Z ( = k^4 * (k^2 - 1) / 4 if n = 2*k + 1). - Michael Somos, Nov 02 2014
a(n) = binomial(n/2,2)^2, n even; a(n) = binomial((n-1)/2,2)*binomial((n+1)/2,2), n odd. - Enrique Navarrete, Dec 22 2019
E.g.f.: (1/128)*exp(-x)*(exp(2*x)*(9 - 12*x + 8*x^2 - 4*x^3 + 2*x^4) - 9 - 6*x - 2*x^2). - Stefano Spezia, Dec 27 2019
a(n) = A002620(n-1)*A002620(n-3)/4. - R. J. Mathar, Mar 23 2021
a(n)= A096338(n-6)+A096338(n-5)+A096338(n-4). - R. J. Mathar, Mar 23 2021
From Amiram Eldar, Mar 20 2022: (Start)
Sum_{n>=5} 1/a(n) = 2*Pi^2/3 - 5.
Sum_{n>=5} (-1)^(n+1)/a(n) = 2*Pi^2 - 19. (End)

A030179 Quarter-squares squared: A002620^2.

Original entry on oeis.org

0, 0, 1, 4, 16, 36, 81, 144, 256, 400, 625, 900, 1296, 1764, 2401, 3136, 4096, 5184, 6561, 8100, 10000, 12100, 14641, 17424, 20736, 24336, 28561, 33124, 38416, 44100, 50625, 57600, 65536, 73984, 83521, 93636, 104976, 116964

Offset: 0

N. J. A. Sloane, Jan 10 2002

Conjectured to be crossing number of complete bipartite graph K_{n,n}. Known to be true for n <= 7.
If the Zarankiewicz conjecture is true, then a(n) is also the rectilinear crossing number of K_{n,n}. - Eric W. Weisstein, Apr 24 2017
a(n+1) is the number of 4-tuples (w,x,y,z) with all terms in {0,...,n}, and w,x,y+1,z+1 all even. - Clark Kimberling, May 29 2012

C. Thomassen, Embeddings and minors, pp. 301-349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.

G. C. Greubel, Table of n, a(n) for n = 0..1000
G. Xiao, Contfrac.
Eric Weisstein's World of Mathematics, Complete Bipartite Graph.
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.
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A000241, A002620, A014540.

List([0..40], n-> (2*n^4 -2*n^2 +1 +(-1)^n*(2*n^2 -1))/32); # G. C. Greubel, Dec 28 2019

[(Floor(n^2/4))^2: n in [0..40]]; // G. C. Greubel, Dec 28 2019

seq( (2*n^4 -2*n^2 +1 +(-1)^n*(2*n^2 -1))/32, n=0..40); # G. C. Greubel, Dec 28 2019

f[n_]:=Floor[n^2/2]; Table[Nest[f,n,2],{n,5!}]/2 (* Vladimir Joseph Stephan Orlovsky, Mar 10 2010 *)
LinearRecurrence[{2,2,-6,0,6,-2,-2,1}, {0,0,1,4,16,36,81,144}, 40] (* Harvey P. Dale, Apr 26 2011 *)
Floor[Range[0, 30]^2/4]^2 (* Eric W. Weisstein, Apr 24 2017 *)

a(n) = (n^2\4)^2 \\ Charles R Greathouse IV, Jun 11 2015

[floor(n^2/4)^2 for n in (0..40)] # G. C. Greubel, Dec 28 2019

a(n) = floor(n^2/4)^2.
From R. J. Mathar, Jul 08 2010: (Start)
G.f.: x^2*(1+2*x+6*x^2+2*x^3+x^4) / ( (1+x)^3*(1-x)^5 ).
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). (End)
a(n) = (2*n^4 -2*n^2 +1 +(-1)^n*(2*n^2 -1))/32. - Luce ETIENNE, Aug 11 2014
Sum_{n>=2} 1/a(n) = Pi^4/90 + Pi^2/3 - 3. - Amiram Eldar, Sep 17 2023

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

A053873 Numbers n such that OEIS sequence A_n contains n.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 14, 16, 19, 26, 27, 36, 37, 52, 59, 62, 69, 72, 115, 119, 120, 121, 134, 161, 164, 174, 177, 188, 189, 190, 193, 194, 195, 196, 209, 224, 265, 267, 277

Offset: 1

Jens Voß, Mar 30 2000

A number n is in this sequence iff n appears anywhere in the terms of A_n, not just in the terms that are visible in the entry.
Is 53873 in this sequence? (A rhetorical question!) - Tanya Khovanova, Aug 09 2007
Is 53169 in this sequence? (A rhetorical question!). - Raymond Wang, Oct 07 2008
I skipped 241 since it appears that A000241(14) > 241, but as the 13th and further terms are not known this is not certain. The next term in the sequence is almost surely 319, but finding the least k for which A000319(k) = 319 requires calculating a chaotic sequence to high precision. - Charles R Greathouse IV, Jul 20 2007
241 is not in this sequence, since A000241(13) <= 225 and A000241(14) >= 0.8594*315 (see comments in A000241). - Danny Rorabaugh, Mar 13 2015

			4 is not in A000004, so 4 is not in this sequence.
60 is not in A000060, so 60 is not in this sequence.
86 is not in A000086, so 86 is not in this sequence.

Charles R. Greathouse IV, Illustration of initial terms
Eric Weisstein's World of Mathematics, Graph Crossing Number
Wikipedia, Self-referentiality in the OEIS
Index entries for sequences whose definition involves A_n (or An).

Complement of A053169.

More terms from N. J. A. Sloane, Aug 24 2006
a(23)-a(25) from Charles R Greathouse IV, Aug 30 2006
a(26)-a(40) from Charles R Greathouse IV, Jul 20 2007
Typo in one entry corrected by Olaf Voß, Feb 25 2008

A007333 An upper bound on the biplanar crossing number of the complete graph on n nodes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 4, 7, 12, 18, 37, 53, 75, 100, 152, 198, 256, 320, 430, 530, 650, 780, 980, 1165, 1380, 1610, 1939, 2247, 2597, 2968, 3472, 3948, 4480, 5040, 5772, 6468, 7236, 8040, 9060, 10035, 11100, 12210, 13585, 14905, 16335, 17820, 19624, 21362

Offset: 1

N. J. A. Sloane

This bound in based on a particular decomposition of K_n (see Owens for details). The actual biplanar crossing number for K_9 is 1 (not 4 as given by this bound). - Sean A. Irvine, Dec 30 2019

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Colin Barker, Table of n, a(n) for n = 1..1000
A. Owens, On the biplanar crossing number, IEEE Trans. Circuit Theory, 18 (1971), 277-280.
A. Owens, On the biplanar crossing number, IEEE Trans. Circuit Theory, 18 (1971), 277-280. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,3,-6,3,0,-3,6,-3,0,1,-2,1).

Cf. A000241, A028723.

LinearRecurrence[{2,-1,0,3,-6,3,0,-3,6,-3,0,1,-2,1},{0,0,0,0,0,0,0,0,4,7,12,18,37,53},70] (* Harvey P. Dale, Feb 13 2022 *)

concat([0,0,0,0,0,0,0,0], Vec(x^9*(4 - x + 2*x^2 + x^3 + x^4) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^3) + O(x^40))) \\ Colin Barker, Feb 02 2020

a(4*k) = k * (k-1) * (k-2) * (7*k-3) / 6, a(4*k+1) = k * (k-1) * (7*k^2-10*k+4) / 6, a(4*k+2) = k * (k-1) * (7*k^2-3*k-1) / 6, a(4*k+3) = k^2 * (k-1) * (7*k+4) / 6 [from Owens]. - Sean A. Irvine, Dec 30 2019; [typo corrected by Colin Barker, Feb 01 2020]
From Colin Barker, Jan 28 2020: (Start)
G.f.: x^9*(4 - x + 2*x^2 + x^3 + x^4) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^3).
a(n) = 2*a(n-1) - a(n-2) + 3*a(n-4) - 6*a(n-5) + 3*a(n-6) - 3*a(n-8) + 6*a(n-9) - 3*a(n-10) + a(n-12) - 2*a(n-13) + a(n-14) for n>14.
(End)

More terms and title clarified by Sean A. Irvine, Dec 30 2019

A145118 Denominator polynomials for continued fraction generating function for n!.

Original entry on oeis.org

1, 1, 1, -1, 1, -2, 1, -4, 2, 1, -6, 6, 1, -9, 18, -6, 1, -12, 36, -24, 1, -16, 72, -96, 24, 1, -20, 120, -240, 120, 1, -25, 200, -600, 600, -120, 1, -30, 300, -1200, 1800, -720, 1, -36, 450, -2400, 5400, -4320, 720, 1, -42, 630, -4200, 12600, -15120

Offset: 0

Paul Barry, Oct 02 2008

Row sums are A056920. T(n,1) gives quarter squares A002620. T(n,2) appears to coincide with 2*A000241(n+1).

			Triangle begins:
1;
1;
1,  -1;
1,  -2;
1,  -4,   2;
1,  -6,   6;
1,  -9,  18,    -6;
1, -12,  36,   -24;
1, -16,  72,   -96,   24;
1, -20, 120,  -240,  120;
1, -25, 200,  -600,  600,  -120;
1, -30, 300, -1200, 1800,  -720;
1, -36, 450, -2400, 5400, -4320, 720;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A021010, A008297, A066667, A105278.

T:= (n, k)-> (-1)^k* binomial(iquo(n+1, 2),k) *binomial(iquo(n, 2), k)*k!:
seq (seq (T(n, k), k=0..iquo(n, 2)), n=0..16);  # Alois P. Heinz, Dec 04 2012

T(n,k) = (-1)^k C(floor((n+1)/2),k) * C(floor(n/2),k)*k!.

A191928 Array read by antidiagonals: T(m,n) = floor(m/2)floor((m-1)/2)floor(n/2)*floor((n-1)/2).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 4, 4, 4, 0, 0, 0, 0, 0, 0, 6, 8, 8, 6, 0, 0, 0, 0, 0, 0, 9, 12, 16, 12, 9, 0, 0, 0, 0, 0, 0, 12, 18, 24, 24, 18, 12, 0, 0, 0, 0, 0, 0, 16, 24, 36, 36, 36, 24, 16, 0, 0, 0, 0, 0, 0, 20, 32, 48, 54, 54, 48, 32, 20, 0, 0, 0, 0, 0, 0, 25, 40, 64, 72, 81, 72, 64, 40, 25, 0, 0, 0

Offset: 0

N. J. A. Sloane, Jun 19 2011

T(m,n) is conjectured to be the crossing number of the complete bipartite graph K_{m,n}.

			Array begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 1, 2, 4, 6, 9, 12, ...
0, 0, 0, 2, 4, 8, 12, 18, 24, ...
0, 0, 0, 4, 8, 16, 24, 36, 48, ...
0, 0, 0, 6, 12, 24, 36, 54, 72, ...
0, 0, 0, 9, 18, 36, 54, 81, 108, ...
0, 0, 0, 12, 24, 48, 72, 108, 144, ...

D. McQuillan and R. B. Richter, A parity theorem for drawings of complete and bipartite graphs, Amer. Math. Monthly, 117 (2010), 267-273.

Cf. A000241, A002620.

K:=(m,n)->floor(m/2)*floor((m-1)/2)*floor(n/2)*floor((n-1)/2);

T(n,k) = ((n-1)^2\4)*((k-1)^2\4);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print()); \\ Michel Marcus, Sep 30 2017

T(m,n) = A002620(m-1)*A002620(n-1). - Michel Marcus, Sep 30 2017

cp's OEIS Frontend

A307891 Rectilinear crossing number A014540(n) - crossing number A000241(n) of complete graph on n nodes.

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A028723 a(n) = (1/4)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2)*floor((n-3)/2).

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A030179 Quarter-squares squared: A002620^2.

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A014540 Rectilinear crossing number of complete graph on n nodes.

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A053873 Numbers n such that OEIS sequence A_n contains n.

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A007333 An upper bound on the biplanar crossing number of the complete graph on n nodes.

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A145118 Denominator polynomials for continued fraction generating function for n!.

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A028723 a(n) = (1/4)floor(n/2)floor((n-1)/2)floor((n-2)/2)floor((n-3)/2).

A191928 Array read by antidiagonals: T(m,n) = floor(m/2)floor((m-1)/2)floor(n/2)*floor((n-1)/2).