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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)

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

Cf. A000241, A028723.

Sequence in context: A178907 A265431 A132297 * A097536 A293829 A344421

Adjacent sequences: A007330 A007331 A007332 * A007334 A007335 A007336

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms and title clarified by Sean A. Irvine, Dec 30 2019

STATUS

approved

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Last modified November 29 16:39 EST 2022. Contains 358431 sequences. (Running on oeis4.)