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 A007333 An upper bound on the biplanar crossing number of the complete graph on n nodes. (Formerly M3306) 2
 0, 0, 0, 0, 0, 0, 0, 0, 4, 7, 12, 18, 37, 53, 75, 100, 152, 198, 256, 320, 430, 530, 650, 780, 980, 1165, 1380, 1610, 1939, 2247, 2597, 2968, 3472, 3948, 4480, 5040, 5772, 6468, 7236, 8040, 9060, 10035, 11100, 12210, 13585, 14905, 16335, 17820, 19624, 21362 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 COMMENTS 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 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Colin Barker, Table of n, a(n) for n = 1..1000 A. Owens, On the biplanar crossing number, IEEE Trans. Circuit Theory, 18 (1971), 277-280. A. Owens, On the biplanar crossing number, IEEE Trans. Circuit Theory, 18 (1971), 277-280. [Annotated scanned copy] Index entries for linear recurrences with constant coefficients, signature (2,-1,0,3,-6,3,0,-3,6,-3,0,1,-2,1). FORMULA 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] From Colin Barker, Jan 28 2020: (Start) G.f.: x^9*(4 - x + 2*x^2 + x^3 + x^4) / ((1 - x)^5*(1 + x)^3*(1 + x^2)^3). 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. (End) PROG (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 CROSSREFS Cf. A000241, A028723. Sequence in context: A178907 A265431 A132297 * A097536 A293829 A022809 Adjacent sequences:  A007330 A007331 A007332 * A007334 A007335 A007336 KEYWORD nonn,nice,easy AUTHOR EXTENSIONS More terms and title clarified by Sean A. Irvine, Dec 30 2019 STATUS approved

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Last modified March 30 21:56 EDT 2020. Contains 333132 sequences. (Running on oeis4.)