A297662 Number of chordless cycles in the complete tripartite graph K_{n,n,n}.

0, 3, 27, 108, 300, 675, 1323, 2352, 3888, 6075, 9075, 13068, 18252, 24843, 33075, 43200, 55488, 70227, 87723, 108300, 132300, 160083, 192027, 228528, 270000, 316875, 369603, 428652, 494508, 567675, 648675, 738048, 836352, 944163, 1062075, 1190700

Offset: 1

Author

Eric W. Weisstein, Jan 02 2018

Keywords

• nonn
• easy

Comments

The only chordless cycles in a complete tripartite graph are cycles of length 4 confined to two of the partitions. - Andrew Howroyd, Jan 03 2018

Links

• Eric Weisstein's World of Mathematics, Chordless Cycle
• Eric Weisstein's World of Mathematics, Complete Tripartite Graph
• Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Crossrefs

Cf. A000537, A234616.

Programs

Mathematica

Table[3 Binomial[n, 2]^2, {n, 20}]
3 Binomial[Range[20], 2]^2
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 3, 27, 108, 300}, 20]
SeriesCoefficient[Series[-3 x (1 + 4 x + x^2)/(-1 + x)^5, {x, 0, 20}], x]

PARI

a(n) = 3*n^2*(n-1)^2/4; \\ Andrew Howroyd, Jan 03 2018

Formula

a(n) = 3*n^2*(n-1)^2/4 = 3*A000537(n). - Andrew Howroyd, Jan 03 2018

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

G.f.: -3*x^2*(1 + 4*x + x^2)/(-1 + x)^5.

Extensions

a(6)-a(36) from Andrew Howroyd, Jan 03 2018