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%I M3306 #34 Feb 13 2022 17:45:48
%S 0,0,0,0,0,0,0,0,4,7,12,18,37,53,75,100,152,198,256,320,430,530,650,
%T 780,980,1165,1380,1610,1939,2247,2597,2968,3472,3948,4480,5040,5772,
%U 6468,7236,8040,9060,10035,11100,12210,13585,14905,16335,17820,19624,21362
%N An upper bound on the biplanar crossing number of the complete graph on n nodes.
%C This bound in based on a particular decomposition of K_n (see Owens for details). The actual biplanar crossing number for K_9 is 1 (not 4 as given by this bound). - _Sean A. Irvine_, Dec 30 2019
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Colin Barker, <a href="/A007333/b007333.txt">Table of n, a(n) for n = 1..1000</a>
%H A. Owens, <a href="https://doi.org/10.1109/TCT.1971.1083266">On the biplanar crossing number</a>, IEEE Trans. Circuit Theory, 18 (1971), 277-280.
%H A. Owens, <a href="/A007333/a007333.pdf">On the biplanar crossing number</a>, IEEE Trans. Circuit Theory, 18 (1971), 277-280. [Annotated scanned copy]
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,0,3,-6,3,0,-3,6,-3,0,1,-2,1).
%F a(4*k) = k * (k-1) * (k-2) * (7*k-3) / 6, a(4*k+1) = k * (k-1) * (7*k^2-10*k+4) / 6, a(4*k+2) = k * (k-1) * (7*k^2-3*k-1) / 6, a(4*k+3) = k^2 * (k-1) * (7*k+4) / 6 [from Owens]. - _Sean A. Irvine_, Dec 30 2019; [typo corrected by _Colin Barker_, Feb 01 2020]
%F From _Colin Barker_, Jan 28 2020: (Start)
%F G.f.: x^9*(4 - x + 2*x^2 + x^3 + x^4) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^3).
%F a(n) = 2*a(n-1) - a(n-2) + 3*a(n-4) - 6*a(n-5) + 3*a(n-6) - 3*a(n-8) + 6*a(n-9) - 3*a(n-10) + a(n-12) - 2*a(n-13) + a(n-14) for n>14.
%F (End)
%t LinearRecurrence[{2,-1,0,3,-6,3,0,-3,6,-3,0,1,-2,1},{0,0,0,0,0,0,0,0,4,7,12,18,37,53},70] (* _Harvey P. Dale_, Feb 13 2022 *)
%o (PARI) concat([0,0,0,0,0,0,0,0], Vec(x^9*(4 - x + 2*x^2 + x^3 + x^4) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^3) + O(x^40))) \\ _Colin Barker_, Feb 02 2020
%Y Cf. A000241, A028723.
%K nonn,nice,easy
%O 1,9
%A _N. J. A. Sloane_
%E More terms and title clarified by _Sean A. Irvine_, Dec 30 2019
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