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.

A231091 Number of distinct (modulo rotation) unicursal star polygons (not necessarily regular, no edge joins adjacent vertices) that can be formed by connecting the vertices of a regular n-gon.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 5, 27, 175, 1533, 14361, 151575, 1735869, 21594863, 289365383, 4158887007, 63822480809, 1041820050629, 18027531255745, 329658402237171, 6352776451924233, 128686951765990343, 2733851297673484765, 60781108703102022027, 1411481990523638719737
Offset: 1

Views

Author

Stewart Gordon, Nov 03 2013

Keywords

Comments

For polygons in general see A000939 and A000949, and especially the Golomb-Welch reference. - N. J. A. Sloane, Nov 21 2013

Examples

			For n=5, only solution is the regular pentagram.
For n=6, only solution is the unicursal hexagram (see Wikipedia link).
For n=7, two regular heptagrams and three irregular forms are possible.

Links

Crossrefs

Cf. A000939 (if edges may join adjacent vertices), A000940, A002816 (rotations and reflections counted separately), A326411, A370459 (up to rotations and reflections), A370068 (directed edges).
Cf. A283184.

Programs

  • PARI
    \\ Requires a370068 from A370068, b(n) is A283184.
b(n)={subst(serlaplace(polcoef((1 - x)/(1 + (1 - 2*y)*x + 2*y*x^2) + O(x*x^n), n)), y, 1)}
a(n)={(if(n%2==0 && n > 2, b(n/2-1)/2) + a370068(n))/2} \\ Andrew Howroyd, Mar 01 2024

Formula

a(n) = (A370068(n) + A283184(n/2-1)/2)/2 for even n >= 4; a(n) = A370068(n)/2 for odd n. - Andrew Howroyd, Feb 24 2024

Extensions

a(15) onwards from Andrew Howroyd, Feb 23 2024

A370068 Number of nonequivalent directed unicursal star polygons (no edge joins adjacent vertices) that can be formed by connecting the vertices of a regular n-gon up to rotations.

Original entry on oeis.org

0, 0, 0, 0, 2, 1, 10, 47, 350, 3005, 28722, 302519, 3471738, 43181993, 578730766, 8317664191, 127644961618, 2083638325661, 36055062511490, 659316772258655, 12705552903848466, 257373902883624297, 5467702595346969530, 121562217391867941767
Offset: 1

Views

Author

Andrew Howroyd, Feb 23 2024

Keywords

Comments

Directed means that the direction of travel is significant.

Links

Crossrefs

Cf. A002619 (if edges may join adjacent vertices), A231091 (undirected), A326411, A370459.

Programs

  • PARI
    Q(n,k)={subst(serlaplace(polcoef((1 - 2*x - x^2)/((1 + x)*(1 + (1 - y)*x + y*x^2)) + O(x^n), n-1)), y, k)}
E(r,d)={eulerphi(d)*Q(r,d) + 2*(-1)^r}
a370068(n)={if(n<3, 0, sumdiv(n,d,eulerphi(d)*E(n/d,d))/n)}

A370769 Number of achiral unicursal star polygons (no edge joins adjacent vertices) that can be formed by connecting the vertices of a regular n-gon.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 5, 11, 49, 123, 521, 1583, 6581, 23239, 95509, 384771, 1570265, 7106995, 28869825, 145034327, 587270877, 3242792607, 13100475021, 78866628011, 318067071169, 2073381189259, 8350998470777, 58602568320255, 235794888434053, 1772311322357623
Offset: 1

Views

Author

Andrew Howroyd, Mar 01 2024

Keywords

Comments

Achiral means that the polygon has an axis of reflective symmetry.

Links

Crossrefs

Cf. A283184, A370766, A370767, A370768.
Cf. A231091 (stars up to rotation), A370459 (up to rotation and reflection).

Programs

  • PARI
    Ro(n)=-(-1)^n + subst(serlaplace(polcoef(((1 - x)^2)/(2*(1 + x)*(1 + (1 - 2*y)*x + 2*y*x^2)) + O(x*x^n), n)), y, 1)
Re(n)=subst(serlaplace(polcoef((1 - 3*x)/(8*(1 + (1 - 2*y)*x + 2*y*x^2)) + O(x*x^n), n)), y, 1)
a(n) = if(n < 3, 0, if(n % 2, Ro(n\2), Re(n/2)))

Formula

a(2*n+1) = A370766(n)/2 - A370768(n-1) for n >= 1.
a(2*n) = (A370766(n-1)/2 - A370768(n-2) + A370766(n)/4 - A370768(n-1) + A283184(n-1)/2)/2 for n >= 2.
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.