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A030179
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Quarter-squares squared: A002620^2.
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20
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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
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OFFSET
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0,4
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COMMENTS
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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
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REFERENCES
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C. Thomassen, Embeddings and minors, pp. 301-349 of R. L. Graham et al., eds., Handbook of Combinatorics, MIT Press.
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LINKS
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FORMULA
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a(n) = floor(n^2/4)^2.
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
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MAPLE
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seq( (2*n^4 -2*n^2 +1 +(-1)^n*(2*n^2 -1))/32, n=0..40); # G. C. Greubel, Dec 28 2019
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MATHEMATICA
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LinearRecurrence[{2, 2, -6, 0, 6, -2, -2, 1}, {0, 0, 1, 4, 16, 36, 81, 144}, 40] (* Harvey P. Dale, Apr 26 2011 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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