A030179 Quarter-squares squared: A002620^2.

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

0,4

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

Cf. A000241, A002620, A014540.

Sequence in context: A063755 A166721 A085040 * A207025 A207170 A207069

Adjacent sequences: A030176 A030177 A030178 * A030180 A030181 A030182

Keyword

nonn,easy

Author

N. J. A. Sloane, Jan 10 2002

Status

approved