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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Examples

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

References

• 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.

Links

• 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).

Crossrefs

Cf. A000241, A006918.

Programs

Magma

[(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

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

SageMath

[(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

Formula

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)