A030179 - OEIS

%I #50 Sep 17 2023 10:01:39

%S 0,0,1,4,16,36,81,144,256,400,625,900,1296,1764,2401,3136,4096,5184,

%T 6561,8100,10000,12100,14641,17424,20736,24336,28561,33124,38416,

%U 44100,50625,57600,65536,73984,83521,93636,104976,116964

%N Quarter-squares squared: A002620^2.

%C Conjectured to be crossing number of complete bipartite graph K_{n,n}. Known to be true for n <= 7.

%C 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

%C 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

%D C. Thomassen, Embeddings and minors, pp. 301-349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.

%H G. C. Greubel, <a href="/A030179/b030179.txt">Table of n, a(n) for n = 0..1000</a>

%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCrossingNumber.html">Graph Crossing Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RectilinearCrossingNumber.html">Rectilinear Crossing Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ZarankiewiczsConjecture.html">Zarankiewicz's Conjecture</a>.

%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 a(n) = floor(n^2/4)^2.

%F From _R. J. Mathar_, Jul 08 2010: (Start)

%F G.f.: x^2*(1+2*x+6*x^2+2*x^3+x^4) / ( (1+x)^3*(1-x)^5 ).

%F 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)

%F a(n) = (2*n^4 -2*n^2 +1 +(-1)^n*(2*n^2 -1))/32. - _Luce ETIENNE_, Aug 11 2014

%F Sum_{n>=2} 1/a(n) = Pi^4/90 + Pi^2/3 - 3. - _Amiram Eldar_, Sep 17 2023

%p seq( (2*n^4 -2*n^2 +1 +(-1)^n*(2*n^2 -1))/32, n=0..40); # _G. C. Greubel_, Dec 28 2019

%t f[n_]:=Floor[n^2/2]; Table[Nest[f,n,2],{n,5!}]/2 (* _Vladimir Joseph Stephan Orlovsky_, Mar 10 2010 *)

%t LinearRecurrence[{2,2,-6,0,6,-2,-2,1}, {0,0,1,4,16,36,81,144}, 40] (* _Harvey P. Dale_, Apr 26 2011 *)

%t Floor[Range[0, 30]^2/4]^2 (* _Eric W. Weisstein_, Apr 24 2017 *)

%o (PARI) a(n) = (n^2\4)^2 \\ _Charles R Greathouse IV_, Jun 11 2015

%o (Magma) [(Floor(n^2/4))^2: n in [0..40]]; // _G. C. Greubel_, Dec 28 2019

%o (Sage) [floor(n^2/4)^2 for n in (0..40)] # _G. C. Greubel_, Dec 28 2019

%o (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

%Y Cf. A000241, A002620, A014540.

%K nonn,easy

%O 0,4

%A _N. J. A. Sloane_, Jan 10 2002