A055684 Number of different n-pointed stars.

0, 0, 1, 0, 2, 1, 2, 1, 4, 1, 5, 2, 3, 3, 7, 2, 8, 3, 5, 4, 10, 3, 9, 5, 8, 5, 13, 3, 14, 7, 9, 7, 11, 5, 17, 8, 11, 7, 19, 5, 20, 9, 11, 10, 22, 7, 20, 9, 15, 11, 25, 8, 19, 11, 17, 13, 28, 7, 29, 14, 17, 15, 23, 9, 32, 15, 21, 11, 34, 11, 35, 17, 19, 17, 29, 11

Offset

3,5

Comments

Does not count rotations or reflections.

This is also the distinct ways of writing a number as the sum of two positive integers greater than one that are coprimes. - Lei Zhou, Mar 19 2014

Equivalently, a(n) is the number of relatively prime 2-part partitions of n without 1's. The Heinz numbers of these partitions are the intersection of A001358 (pairs), A005408 (no 1's), and A000837 (relatively prime) or A302696 (pairwise coprime). - Gus Wiseman, Oct 28 2020

References

Mark A. Herkommer, "Number Theory, A Programmer's Guide," McGraw-Hill, New York, 1999, page 58.

Links

Lei Zhou, Table of n, a(n) for n = 3..10002

Alexander Bogomolny, Polygons: formality and intuition.. Includes applet to draw star polygons.

Vi Hart, Doodling in Math Class: Stars, Video (2010).

Hugo Pfoertner, Star-shaped regular polygons up to n=25.

Eric Weisstein's World of Mathematics, Star Polygon

Formula

a(n) = A023022(n) - 1.

a(n) + A082023(n) = A140106(n). - Gus Wiseman, Oct 28 2020

Example

The first star has five points and is unique. The next is the seven pointed star and it comes in two varieties.
From Gus Wiseman, Oct 28 2020: (Start)
The a(5) = 1 through a(17) = 7 irreducible pairs > 1 (shown as fractions, empty column indicated by dot):
  2/3  .  2/5  3/5  2/7  3/7  2/9  5/7  2/11  3/11  2/13  3/13  2/15
          3/4       4/5       3/8       3/10  5/9   4/11  5/11  3/14
                              4/7       4/9         7/8   7/9   4/13
                              5/6       5/8                     5/12
                                        6/7                     6/11
                                                                7/10
                                                                8/9
(End)

Maple

with(numtheory): A055684 := n->(phi(n)-2)/2; seq(A055684(n), n=3..100);

Mathematica

Table[(EulerPhi[n]-2)/2, {n, 3, 50}]
Table[Length[Select[IntegerPartitions[n, {2}], !MemberQ[#, 1]&&CoprimeQ@@#&]], {n, 0, 30}] (* Gus Wiseman, Oct 28 2020 *)

Crossrefs

Cf. A023022.

Cf. A053669 smallest skip increment, A102302 skip increment of densest star polygon.

A055684*2 is the ordered version.

A082023 counts the complement (reducible pairs > 1).

A220377, A337563, and A338332 count triples instead of pairs.

A000837 counts relatively prime partitions, with strict case A078374.

A002865 counts partitions with no 1's, with strict case A025147.

A007359 and A337485 count pairwise coprime partitions with no 1's.

A302698 counts relatively prime partitions with no 1's, with strict case A337452.

A327516 counts pairwise coprime partitions, with strict case A305713.

A337450 counts relatively prime compositions with no 1's, with strict case A337451.

Cf. A001399, A101268, A140106, A337461, A337462, A338333.

Sequence in context: A231473 A216652 A331980 * A300584 A024559 A334664

Adjacent sequences: A055681 A055682 A055683 * A055685 A055686 A055687

Keyword

nonn,easy

Author

Robert G. Wilson v, Jun 09 2000

Status

approved