OFFSET
0,7
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
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).
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
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
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)
EXAMPLE
G.f. = x^5 + 3*x^6 + 9*x^7 + 18*x^8 + 36*x^9 + 60*x^10 + 100*x^11 + ...
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 *)
PROG
(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 */
(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
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved