This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A007678 Number of regions in regular n-gon with all diagonals drawn. (Formerly M3411) 33
 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 Also the circuit rank of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 08 2018 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 M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5 Sascha Kurz, m-gons in regular n-gons J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975 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 M. Rubinstein, Drawings for n=4,5,6,... Eric Weisstein's World of Mathematics, Circuit Rank Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals 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.

Last modified November 17 04:01 EST 2018. Contains 317275 sequences. (Running on oeis4.)