A319127 Crossing number of the complete bipartite graph K_{6,n}.

0, 0, 6, 12, 24, 36, 54, 72, 96, 120, 150, 180, 216, 252, 294, 336, 384, 432, 486, 540, 600, 660, 726, 792, 864, 936, 1014, 1092, 1176, 1260, 1350, 1440, 1536, 1632, 1734, 1836, 1944, 2052, 2166, 2280, 2400, 2520, 2646, 2772, 2904, 3036, 3174, 3312, 3456, 3600, 3750

Offset: 1

Author

Eric W. Weisstein, Sep 11 2018

Keywords

• nonn
• easy

Links

• Eric Weisstein's World of Mathematics, Complete Bipartite Graph
• Eric Weisstein's World of Mathematics, Graph Crossing Number
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Programs

Mathematica

Table[6 Floor[n/2] Floor[(n - 1)/2], {n, 60}]
Table[3/4 (2 n (n - 2) + 1 - (-1)^n), {n, 60}]
LinearRecurrence[{2, 0, -2, 1}, {0, 0, 6, 12}, 60]
CoefficientList[Series[-6 x^2/((-1 + x)^3 (1 + x)), {x, 0, 60}], x]

PARI

a(n)=n--^2\4*6 \\ Charles R Greathouse IV, Jul 13 2021

Formula

a(n) = 6*floor(n/2)*floor((n-1)/2).

G.f.: -6*x^3/((-1 + x)^3*(1 + x)).

a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).

a(n) = (3/4)*(2*n*(n - 2) + 1 - (-1)^n).

a(n) = 6*A002620(n-1). - R. J. Mathar, Feb 12 2021