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.

A332426 Number of unordered pairs of non-selfintersecting paths with nodes that cover all vertices of a convex n-gon.

Original entry on oeis.org

0, 3, 30, 210, 1260, 6944, 36288, 182880, 897600, 4316928, 20427264, 95373824, 440294400, 2013020160, 9126248448, 41069371392, 183607050240, 816037560320, 3607758766080, 15874168848384, 69544044134400, 303465064562688
Offset: 3

Views

Author

Ivaylo Kortezov, Feb 12 2020

Keywords

Comments

Although each path is non-selfintersecting, the two paths are allowed to intersect.
Paths must have at least two nodes.
The number of non-selfintersecting paths that cover all vertices of a convex n-gon is given by A001792(n-2).
Given a sequence of two or more different vertices of the n-gon, if we connect each vertex after the first one by a segment to the preceding vertex, then the union of these segments is a path (the direction does not matter).

Examples

			a(5)=30 since one of the paths has to be a segment connecting two vertices (10 choices) and the other path will connect the remaining vertices in one of three ways.

Links

Crossrefs

Cf. A001792, A308914.

Programs

  • Maple
    gf := ((exp(2*x)-1)*x)^2/32: ser := series(gf, x, 32):
seq(n!*coeff(ser, x, n), n=3..24); # Peter Luschny, Mar 01 2020

Formula

a(n) = n*(n-1)*2^(n-6)*(2^(n-3)-1).
a(n) = (1/2)*Sum_{k=2..n-2} binomial(n,k)*A001792(k-2)*A001792(n-k-2).
From Stefano Spezia, Feb 12 2020: (Start)
O.g.f.: x^4*(3 - 24*x + 66*x^2 - 72*x^3 + 32*x^4)/(1 - 6*x + 8*x^2)^3.
E.g.f.: (exp(2*x) - 1)^2*x^2/32.
a(n) = 18*a(n-1) - 132*a(n-2) + 504*a(n-3) - 1056*a(n-4) + 1152*a(n-5) - 512*a(n-6) for n > 8.
(End)

A362786 Number of unordered triples of disjoint self-avoiding paths with nodes that cover all vertices of a convex n-gon.

Original entry on oeis.org

0, 0, 0, 5, 63, 476, 2772, 13680, 60060, 241472, 906048, 3214848, 10890880, 35481600, 111794176, 342171648, 1021031424, 2979102720, 8520171520, 23934468096, 66156625920, 180198047744, 484304486400, 1285790105600, 3375480176640, 8769899593728, 22567515586560, 57557594931200
Offset: 3

Views

Author

Ivaylo Kortezov, May 04 2023

Keywords

Examples

			For n=7 we have one 3-node path and two 2-node paths. Call two paths adjacent if we can choose one node from each path so that the two nodes are adjacent vertices of the n-gon. Then either each pair of paths is adjacent, or the two 2-node paths are not adjacent, or a 2-node path is not adjacent to the 3-node path. In each of these three cases there are 7 choices for the set of nodes for the 3-node path and 3 ways to connect them, and then the 2-node paths are uniquely determined. Thus a(7) = 3*7*3 = 63.

Crossrefs

The number of unordered pairs of disjoint self-avoiding paths with nodes that cover all vertices of a convex n-gon is A308914(n). The number of unordered triples of (not necessarily disjoint) self-avoiding paths with nodes that cover all vertices of a convex n-gon is A359404(n).

Formula

a(n) = 2^(n-12)*n*(n-1)*(n-2)*(n-4)*(n-5)*(n+2)*(n+9)/90 for n > 3; 0 for n=3.

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.

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