cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A360717 Number of unordered pairs of self-avoiding paths whose sets of nodes are disjoint subsets of a set of n points on a circle; one-node paths are allowed.

Original entry on oeis.org

0, 1, 6, 33, 185, 1050, 6027, 35014, 205326, 1209375, 7119860, 41744703, 243218703, 1406685280, 8073640785, 45991600860, 260131208396, 1461591509805, 8162196518322, 45327133739245, 250431036147285, 1377169337010390, 7540979990097191, 41130452834689218, 223528009015333050, 1210753768099880875, 6537995998163877312
Offset: 1

Views

Author

Ivaylo Kortezov, Feb 18 2023

Keywords

Comments

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

Examples

			a(4) = A359405(4) + 4*A359405(3) + 4*3/2 = 15 + 12 + 6 = 33 with the three summands corresponding to the cases of 4, 3 and 2 used points.

Links

Crossrefs

If there is only one path, we get A360715. If one-node paths are not allowed, we get A360716.

Formula

a(n) = n*(n-1)*2^(-5)*(5^(n-2) + 6*3^(n-2) + 9).
E.g.f.: exp(x)*((x*exp(2*x) + 3*x)/4)^2/2. - Andrew Howroyd, Feb 19 2023

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.

Original entry on oeis.org

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
Offset: 1

Views

Author

Ivaylo Kortezov, Mar 07 2023

Keywords

Comments

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

Examples

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

Links

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.

Programs

  • 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

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

A363964 Number of unordered pairs of non-intersecting non-self-intersecting paths, singletons included, with nodes that cover all vertices of a convex labeled n-gon.

Original entry on oeis.org

3, 14, 55, 195, 644, 2016, 6048, 17520, 49280, 135168, 362752, 955136, 2472960, 6307840, 15876096, 39481344, 97124352, 236584960, 571146240, 1367539712, 3249799168, 7669284864, 17983078400, 41916825600, 97165246464, 224076496896, 514272002048, 1174992322560
Offset: 3

Views

Author

Ivaylo Kortezov, Jun 30 2023

Keywords

Comments

For each such path there is a sequence of distinct vertices of the n-gon, each (except the last one) connected by a segment with the next vertex in the sequence; the segments have no common internal points. The path itself is the union of the set of these segments and is thus direction-independent: reversing the order of the vertices leads to the same path. If the sequence of vertices has length 1 then there are no segments; we call such a path a singleton.

Examples

			a(4)=14 since if one of the paths is a singleton (4 choices), then there are A001792(3)=3 choices for the other path, and otherwise for the two paths there are A308914(4)=2 choices, so a(4)=4*3+2=14.

Crossrefs

Cf. A001792, A308914, A332426, A360716, A360717.

Formula

a(n) = n*(n-1)*(n^2+n+36)*2^(n-8)/3.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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