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.

A007678 Number of regions in regular n-gon with all diagonals drawn.

Original entry on oeis.org

0, 0, 1, 4, 11, 24, 50, 80, 154, 220, 375, 444, 781, 952, 1456, 1696, 2500, 2466, 4029, 4500, 6175, 6820, 9086, 9024, 12926, 13988, 17875, 19180, 24129, 21480, 31900, 33856, 41416, 43792, 52921, 52956, 66675, 69996, 82954, 86800, 102050, 97734, 124271, 129404, 149941
Offset: 1

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Author

N. J. A. Sloane, Bjorn Poonen (poonen(AT)math.princeton.edu)

Keywords

Comments

This sequence and A006533 are two equivalent ways of presenting the same sequence.
A quasipolynomial of order 2520. - Charles R Greathouse IV, Jan 15 2013
Also the circuit rank of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 08 2018
This sequence only counts polygons, in contrast to A006533 which also counts the n segments of the circumscribed circle delimited by the edges of the regular n-gon. Therefore a(n) = A006533(n) - n. See also A006561 which counts the intersection points, and A350000 which considers iterated "cutting along diagonals". - M. F. Hasler, Dec 13 2021
The Petrie polygon orthographic projection of a regular n-simplex is a regular (n+1)-gon with all diagonals drawn. Hence a(n+1) is the number of regions in the Petrie polygon of a regular n-simplex. - Mohammed Yaseen, Nov 05 2022

References

  • Jean Meeus, Wiskunde Post (Belgium), Vol. 10, 1972, pp. 62-63.
  • C. A. Pickover, The Mathematics of Oz, Problem 58 "The Beauty of Polygon Slicing", Cambridge University Press, 2002.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001006, A054726, A006533, A006561, A006600, A007569 (number of vertices), A006522, A135565 (number of line segments).
A062361 gives number of triangles, A331450 and A331451 give distribution of polygons by number of sides.
A333654, A335614, A335646, A337330 give the number of internal n-gon to k-gon contacts for n>=3, k>=n.
A187781 gives number of distinct regions.

Programs

  • Mathematica
    del[m_,n_]:=If[Mod[n,m]==0,1,0]; R[n_]:=If[n<3, 0, (n^4-6n^3+23n^2-42n+24)/24 + del[2,n](-5n^3+42n^2-40n-48)/48 - del[4,n](3n/4) + del[6,n](-53n^2+310n)/12 + del[12,n](49n/2) + del[18,n]*32n + del[24,n]*19n - del[30,n]*36n - del[42,n]*50n - del[60,n]*190n - del[84,n]*78n - del[90,n]*48n - del[120,n]*78n - del[210,n]*48n]; Table[R[n], {n,1,1000}] (* T. D. Noe, Dec 21 2006 *)
  • PARI
    /* Only for odd n > 3, not suitable for other values of n! */ { a(n)=local(nr,x,fn,cn,fn2); nr=0; fn=floor(n/2); cn=ceil(n/2); fn2=(fn-1)^2-1; nr=fn2*n+fn+(n-2)*fn+cn; x=(n-5)/2; if (x>0,nr+=x*(x+1)*(2*x+1)/6*n); nr; } \\ Jon Perry, Jul 08 2003
  • PARI
    apply( {A007678(n)=if(n%2, (((n-6)*n+23)*n-42)*n/24+1, ((n^3/2 -17*n^2/4 +22*n -if(n%4, 31, 40) +!(n%6)*(310 -53*n))/12 +!(n%12)*49/2 +!(n%18)*32 +!(n%24)*19 -!(n%30)*36 -!(n%42)*50 -!(n%60)*190 -!(n%84)*78 -!(n%90)*48 -!(n%120)*78 -!(n%210)*48)*n)}, [1..44]) \\ M. F. Hasler, Aug 06 2021
  • Python
    def d(n,m): return not n % m
def A007678(n): return (1176*d(n,12)*n - 3744*d(n,120)*n + 1536*d(n,18)*n - d(n,2)*(5*n**3 - 42*n**2 + 40*n + 48) - 2304*d(n,210)*n + 912*d(n,24)*n - 1728*d(n,30)*n - 36*d(n,4)*n - 2400*d(n,42)*n - 4*d(n,6)*n*(53*n - 310) - 9120*d(n,60)*n - 3744*d(n,84)*n - 2304*d(n,90)*n + 2*n**4 - 12*n**3 + 46*n**2 - 84*n)//48 + 1 # Chai Wah Wu, Mar 08 2021

Formula

For odd n > 3, a(n) = sumstep {i=5, n, 2, (i-2)*floor(n/2)+(i-4)*ceiling(n/2)+1} + x*(x+1)*(2*x+1)/6*n), where x = (n-5)/2. Simplifying the floor/ceiling components gives the PARI code below. - Jon Perry, Jul 08 2003
For odd n, a(n) = (24 - 42*n + 23*n^2 - 6*n^3 + n^4)/24. - Graeme McRae, Dec 24 2004
a(n) = A006533(n) - n. - T. D. Noe, Dec 23 2006
For odd n, binomial transform of [1, 10, 29, 36, 16, 0, 0, 0, ...] = [1, 11, 50, 154, ...]. - Gary W. Adamson, Aug 02 2011
a(n) = A135565(n) - A007569(n) + 1. - Max Alekseyev
See the Mma code in A006533 for the explicit Poonen-Rubenstein formula that holds for all n. - N. J. A. Sloane, Jan 23 2020

Extensions

More terms from Graeme McRae, Dec 26 2004
a(1) = a(2) = 0 prepended by Max Alekseyev, Dec 01 2011

A217753 Number of noncongruent polygonal regions with finite area in the exterior of a regular n-gon with all diagonals drawn.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 7, 14, 18, 30, 35, 55, 62, 90
Offset: 3

Views

Author

Martin Renner, Mar 23 2013

Keywords

Examples

			a(7) = 1 since the 35 exterior regions of the regular heptagon built by all diagonals consist of one noncongruent polygon, i.e., 1 triangle (7 times), and three different noncongruent regions with infinite area (two 7 times, one 14 times).
a(8) = 2 since the 64 exterior regions of the regular octagon built by all diagonals consist of two different noncongruent polygons, i.e., 2 triangles (one 8 times, one 16 times), and three different noncongruent regions with infinite area (one 8 times, two 16 times).
a(9) = 5 since the 117 exterior regions of the regular 9-gon (nonagon) built by all diagonals consist of five different noncongruent polygons, i.e., 3 triangles (one 9 times, two 18 times) and 2 quadrilaterals (each 9 times), and four different noncongruent regions with infinite area (two 9 times, two 18 times).

Crossrefs

Cf. A004526, A187781, A217745, A217746, A217748, A217754.

A363979 Number of nonsimilar polygonal regions in a regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 1, 3, 3, 7, 7, 14, 13, 25, 19, 37, 35, 58, 53, 82, 58, 112, 98, 150, 133, 177, 151, 239, 212, 300, 275, 369, 244, 455, 409, 530, 488, 631, 526, 736, 683, 858, 800, 975, 794, 1133, 1056, 1291, 1227, 1487, 1289, 1666, 1600, 1889, 1797, 2102, 1884, 2373, 2242, 2621, 2496, 2878, 2458
Offset: 3

Views

Author

Christopher Scussel, Jun 30 2023

Keywords

Comments

Alternatively, the number of equivalence classes of an equivalence relation on the polygonal regions in a regular n-gon with all diagonals drawn, where two regions are equivalent iff they are similar.

Examples

			This sequence is the same as A187781 for n<10 because n=10 is the smallest n such that there exist regions that are similar but not congruent.

Links

Crossrefs

Cf. A187781, A007678.
Showing 1-3 of 3 results.

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