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A007333
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An upper bound on the biplanar crossing number of the complete graph on n nodes.
(Formerly M3306)
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2
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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
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OFFSET
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1,9
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COMMENTS
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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
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (2,-1,0,3,-6,3,0,-3,6,-3,0,1,-2,1).
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FORMULA
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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]
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)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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