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%I #112 Apr 09 2022 01:27:20
%S 0,0,0,0,0,1,3,9,18,36,60,100,150,225,315,441,588,784,1008,1296,1620,
%T 2025,2475,3025,3630,4356,5148,6084,7098,8281,9555,11025,12600,14400,
%U 16320,18496,20808,23409,26163,29241,32490,36100,39900,44100,48510,53361,58443
%N a(n) = (1/4)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2)*floor((n-3)/2).
%C It is not known whether A000241 and this sequence agree.
%C Conjectured to be crossing number of complete graph K_n, see A000241.
%C a(n+1) is the maximum number of rectangles that can be formed from n lines. - _Erich Friedman_
%C 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
%C 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
%C 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
%C 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
%C 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
%C 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
%D Martin Gardner, Knotted Doughnuts and Other Mathematical Entertainments, W. H. Freeman & Company, 1986, Chapter 11, pages 133-144.
%D Carsten Thomassen, Embeddings and Minors, in: R. L. Graham, M. Grötschel, and L. Lovász, Handbook of Combinatorics, Vol. 1, Elsevier, 1995, p. 314.
%H Vincenzo Librandi, <a href="/A028723/b028723.txt">Table of n, a(n) for n = 0..1000</a>
%H Bernardo M. Abrego, Oswin Aichholzer, Silvia Fernandez-Merchant, Pedro Ramos, and Gelasio Salazar, <a href="https://arxiv.org/abs/1206.5669">The 2-Page Crossing Number of K_n</a>, arXiv:1206.5669 [math.CO], 2012.
%H Bernardo M. Abrego, Oswin Aichholzer, Silvia Fernandez-Merchant, Pedro Ramos, and Gelasio Salazar, <a href="https://doi.org/10.1007/s00454-013-9514-0">The 2-Page Crossing Number of K_n</a>, Discrete Comput. Geom., Vol. 49, No. 4 (2013), pp. 747-777. MR3068573.
%H James Dolan et al., <a href="https://web.archive.org/web/20041126181132/http://mathforum.org/epigone/sci.math.research/stroblequy">su(3) and Zarankiewicz's conjecture</a>.
%H Dhruv Mubayi, <a href="http://dx.doi.org/10.1007/s00493-013-2638-2">Counting substructures II: Hypergraphs</a>, Combinatorica, Vol. 33, No. 5 (2013), pp. 591--612. MR3132928
%H Nicholas Pippenger and Martin Charles Golumbic, <a href="https://doi.org/10.1016/0095-8956(75)90084-2">The inducibility of graphs</a>, Journal of Combinatorial Theory Series B 19 (1975), 189-203.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).
%F If n even, n*(n-2)^2*(n-4)/64; if n odd, (n-1)^2*(n-3)^2/64.
%F G.f.: x^5*(1+x+x^2)/((1-x)^5*(1+x)^3). - _Emeric Deutsch_, Jan 12 2004
%F For n>2, a(n) = A007590(n-3)*A007590(n-1)/16. - _Richard R. Forberg_, Dec 03 2013
%F 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
%F Euler transform of length 3 sequence [3, 3, -1]. - _Michael Somos_, Nov 02 2014
%F a(n) = a(4-n) for all n in Z. - _Michael Somos_, Nov 02 2014
%F 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
%F 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
%F 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
%F 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
%F 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
%F 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
%F a(n) = A002620(n-1)*A002620(n-3)/4. - _R. J. Mathar_, Mar 23 2021
%F a(n)= A096338(n-6)+A096338(n-5)+A096338(n-4). - _R. J. Mathar_, Mar 23 2021
%F From _Amiram Eldar_, Mar 20 2022: (Start)
%F Sum_{n>=5} 1/a(n) = 2*Pi^2/3 - 5.
%F Sum_{n>=5} (-1)^(n+1)/a(n) = 2*Pi^2 - 19. (End)
%e G.f. = x^5 + 3*x^6 + 9*x^7 + 18*x^8 + 36*x^9 + 60*x^10 + 100*x^11 + ...
%p 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
%t Table[If[EvenQ[n], n(n-2)^2(n-4)/64, (n-1)^2(n-3)^2/64], {n, 0, 50}]
%t 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 *)
%t LinearRecurrence[{2, 2,-6,0,6,-2,-2,1}, {0,0,0,0,0,1,3,9}, 50] (* _Harvey P. Dale_, Sep 13 2018 *)
%t Times@@@Table[Floor[(n-k)/2], {n,0,60}, {k,0,3}]/4 (* _Eric W. Weisstein_, Apr 29 2019 *)
%o (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
%o (PARI) {a(n) = prod(k=0, 3, (n - k) \ 2) / 4}; /* _Michael Somos_, Nov 02 2014 */
%o (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
%o (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
%Y Cf. A000241, A006918.
%K nonn,easy
%O 0,7
%A _N. J. A. Sloane_
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