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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
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)
MATHEMATICA
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 *)
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
Sequence in context: A178907 A265431 A132297 * A097536 A293829 A344421
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 28 14:02 EDT 2024. Contains 371254 sequences. (Running on oeis4.)