OFFSET
0,4
COMMENTS
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
REFERENCES
C. Thomassen, Embeddings and minors, pp. 301-349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.
LINKS
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).
FORMULA
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
MAPLE
seq( (2*n^4 -2*n^2 +1 +(-1)^n*(2*n^2 -1))/32, n=0..40); # G. C. Greubel, Dec 28 2019
MATHEMATICA
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 *)
PROG
(PARI) a(n) = (n^2\4)^2 \\ Charles R Greathouse IV, Jun 11 2015
(Magma) [(Floor(n^2/4))^2: n in [0..40]]; // G. C. Greubel, Dec 28 2019
(Sage) [floor(n^2/4)^2 for n in (0..40)] # G. C. Greubel, Dec 28 2019
(GAP) List([0..40], n-> (2*n^4 -2*n^2 +1 +(-1)^n*(2*n^2 -1))/32); # G. C. Greubel, Dec 28 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 10 2002
STATUS
approved