A361284 - OEIS

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A361284

Number of unordered triples of self-avoiding paths whose sets of nodes are disjoint subsets of a set of n points on a circle; one-node paths are not allowed.

0, 0, 0, 0, 0, 15, 420, 7140, 95760, 1116990, 11891880, 118776900, 1132182480, 10415938533, 93207174060, 815777235000, 7011723045600, 59364660734172, 496238466573648, 4102968354298200, 33602671702168800, 272909132004479355, 2200084921469527092, 17618774018675345340, 140252152286127750000

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OFFSET

1,6

COMMENTS

Although each path is self-avoiding, the different paths are allowed to intersect.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..500

Ivaylo Kortezov, Sets of Paths between Vertices of a Polygon, Mathematics Competitions, Vol. 35 (2022), No. 2, ISSN:1031-7503, pp. 35-43.

FORMULA

a(n) = (n*(n-1)*(n-2)/384)*(7^(n-3) - 3*5^(n-3) + 3^(n-2) - 1).

E.g.f.: x^3*exp(x)*(exp(2*x) - 1)^3/384. - Andrew Howroyd, Mar 07 2023

EXAMPLE

a(7) = A359404(7) + 7*A359404(6) = 315 + 7*15 = 420 since either all the 7 points are used or one is not.

PROG

(PARI) a(n) = {(n*(n-1)*(n-2)/384) * (7^(n-3) - 3*5^(n-3) + 3^(n-2) - 1)} \\ Andrew Howroyd, Mar 07 2023

CROSSREFS

If there is only one path, we get A261064. If there is are two paths, we get A360716. If all n points need to be used, we get A359404.

Sequence in context: A184222 A069431 A133791 * A323781 A253447 A302112

Adjacent sequences: A361281 A361282 A361283 * A361285 A361286 A361287

KEYWORD

nonn,easy

AUTHOR

Ivaylo Kortezov, Mar 07 2023

STATUS

approved