A102302 -id:A102302 - cp's OEIS frontend

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

Robert G. Wilson v, Jun 09 2000

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

			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)

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

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

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.

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

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 *)

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

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

A330662 Triangle read by rows: T(n,k) is the number of polygons with 2n sides, of which k run through the center of a circle, on the circumference of which the 2n vertices of the polygon are arranged at equal spacing.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 16, 24, 12, 8, 744, 960, 576, 192, 48, 56256, 69120, 39360, 13440, 2880, 384, 6385920, 7580160, 4204800, 1420800, 316800, 46080, 3840, 1018114560, 1178956800, 642539520, 216115200, 49190400, 7741440, 806400, 46080

Offset: 0

Ludovic Schwob, Dec 23 2019

Rotations and reflections are counted separately.
By "2*n-sided polygons" we mean the polygons that can be drawn by connecting 2*n equally spaced points on a circle.
T(0,0)=0 and T(0,1)=1 by convention.
The sequence is limited to even-sided polygons, since all odd-sided polygons have no side passing through the center.

			Triangle begins:
    0;
    0,   1;
    1,   0,   2;
   16,  24,  12,   8;
  744, 960, 576, 192, 48;

Ludovic Schwob, Table of n, a(n) for n = 0..494
Ludovic Schwob, Illustration of T(3,k), 0≤k≤3.

Row sums give A001710(2*n-1) (number of polygons with 2*n sides).
Cf. A000165 (diagonal).
Star polygons: A014106, A055684, A102302.
Cf. A309318.

T := (n, k) -> `if`(n<2, k, 2^(k-1)*binomial(n,k)*(2*n-k-1)!*hypergeom([k-n], [k-2*n+ 1], -2)):
seq(seq(simplify(T(n,k)), k=0..n),n=0..7); # Peter Luschny, Jan 07 2020

T(n,n) = 2^(n-1) * (n-1)! for all n >= 1.
T(n,0) = A307923(n) for all n>=1.
T(n,k) = binomial(n,k)* Sum_{i=k..n} (-1)^(i-k)*binomial(n-k,i-k)*(2n-1-i)!*2^(i-1), for n>=2 and 0<=k<=n.

A173989 a(n) is the 2-adic valuation of A173300(n).

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 3, 4, 3, 5, 5, 6, 5, 7, 7, 8, 7, 9, 9, 10, 9, 11, 11, 12, 11, 13, 13, 14, 13, 15, 15, 16, 15, 17, 17, 18, 17, 19, 19, 20, 19, 21, 21, 22, 21, 23, 23, 24, 23, 25, 25, 26, 25, 27, 27, 28, 27, 29, 29, 30, 29, 31, 31, 32, 31, 33, 33, 34, 33, 35, 35, 36, 35, 37, 37, 38, 37

Offset: 1

J. Lowell, Mar 04 2010

Conjecture: always follows the pattern A, A, A+1, A, where A is an odd number.

Hugo Pfoertner, Table of n, a(n) for n = 1..1000

Cf. A102302, A173300.

From R. J. Mathar, Mar 20 2010: (Start)
A173300 := proc(n) local x,y ; x := (1+sqrt(3))/2 ; y := (1-sqrt(3))/2 ; denom(expand(x^n+y^n)) ; end proc:
A173989 := proc(n) log[2](A173300(n)) ; end proc: seq(A173989(n),n=3..100) ; (End)

Log2[Denominator[Map[First, NestList[{Last[#], Last[#] + First[#]/2} &, {1, 2}, 100]]]] (* Paolo Xausa, Feb 01 2024, after Nick Hobson in A173300 *)

\\ using Max Alekseyev's function in A173300
A173300(n) = denominator(2*polcoeff( lift( Mod((1+x)/2, x^2-3)^n ), 0))
for(k=1,74,print1(logint(A173300(k),2),", ")) \\ Hugo Pfoertner, Oct 10 2018

a(n) = log(A173300(n))/log(2).
Apparently a(n) = A102302(n) for n >= 7. - Hugo Pfoertner, Oct 10 2018
Conjectures from Colin Barker, Oct 10 2018: (Start)
G.f.: x^3*(1 + x^2 - x^3 + x^4) / ((1 - x)^2*(1 + x)*(1 + x^2)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 7.
(End)
Apparently a(n) = A116921(n) for n>=3. - R. J. Mathar, Aug 29 2025

More terms from R. J. Mathar and Max Alekseyev, Mar 20 2010

cp's OEIS Frontend

A055684 Number of different n-pointed stars.

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A330662 Triangle read by rows: T(n,k) is the number of polygons with 2n sides, of which k run through the center of a circle, on the circumference of which the 2n vertices of the polygon are arranged at equal spacing.

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Maple

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A173989 a(n) is the 2-adic valuation of A173300(n).

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Author

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Programs

Maple

Mathematica

PARI

Formula

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cp's OEIS Frontend

A055684 Number of different n-pointed stars.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A330662 Triangle read by rows: T(n,k) is the number of polygons with 2*n sides, of which k run through the center of a circle, on the circumference of which the 2*n vertices of the polygon are arranged at equal spacing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A173989 a(n) is the 2-adic valuation of A173300(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A330662 Triangle read by rows: T(n,k) is the number of polygons with 2n sides, of which k run through the center of a circle, on the circumference of which the 2n vertices of the polygon are arranged at equal spacing.