login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A007678 Number of regions in regular n-gon with all diagonals drawn.
(Formerly M3411)
32
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Quasipolynomial of order 2520. - Charles R Greathouse IV, Jan 15 2013

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

T. D. Noe, Table of n, a(n) for n = 1..1000

Sascha Kurz, m-gons in regular n-gons

B. Poonen and M. Rubinstein (1998) The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11(1), pp. 135-156, doi:10.1137/S0895480195281246, arXiv:math.MG/9508209 (has fewer typos than the SIAM version)

B. Poonen and M. Rubinstein, Mathematica programs for these sequences

Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals

Sequences formed by drawing all diagonals in regular polygon

Sequences related to chord diagrams

FORMULA

For odd n>3, a(n) = sumstep {i=5, n, 2, (i-2)*floor(n/2)+(i-4)*ceil(n/2)+1} + x*(x+1)*(2*x+1)/6*n), where x=(n-5)/2. Simplifying the floor/ceil components gives the PARI code below. - Jon Perry, Jul 08 2003

For odd n, a(n) = (24 - 42n + 23n^2 - 6n^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. [From Max Alekseyev]

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

PROG

(PARI) { 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; }

CROSSREFS

Cf. A001006, A054726, A006533, A006561, A006600, A007569, A006522.

Sequence in context: A260057 A260150 A258472 * A159350 A159348 A159349

Adjacent sequences:  A007675 A007676 A007677 * A007679 A007680 A007681

KEYWORD

easy,nonn,nice

AUTHOR

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

EXTENSIONS

More terms from Graeme McRae, Dec 26 2004

a(1)=a(2)=0 prepended by Max Alekseyev, Dec 01 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified March 29 18:39 EDT 2017. Contains 284273 sequences.