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 A007678 Number of regions in regular n-gon with all diagonals drawn. (Formerly M3411) 33

%I M3411

%S 0,0,1,4,11,24,50,80,154,220,375,444,781,952,1456,1696,2500,2466,4029,

%T 4500,6175,6820,9086,9024,12926,13988,17875,19180,24129,21480,31900,

%U 33856,41416,43792,52921,52956,66675,69996,82954,86800,102050

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

%C Quasipolynomial of order 2520. - _Charles R Greathouse IV_, Jan 15 2013

%C Also the circuit rank of the n-polygon diagonal intersection graph. - _Eric W. Weisstein_, Mar 08 2018

%D Jean Meeus, Wiskunde Post (Belgium), Vol. 10, 1972, pp. 62-63.

%D C. A. Pickover, The Mathematics of Oz, Problem 58 "The Beauty of Polygon Slicing", Cambridge University Press, 2002.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A007678/b007678.txt">Table of n, a(n) for n = 1..1000</a>

%H M. Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Griffiths2/griffiths.html">Counting the regions in a regular drawing of K_{n,n}</a>, J. Int. Seq. 13 (2010) # 10.8.5

%H Sascha Kurz, <a href="http://www.mathe2.uni-bayreuth.de/sascha/oeis/drawing/drawing.html">m-gons in regular n-gons</a>

%H J. Meeus & N. J. A. Sloane, <a href="/A006532/a006532_1.pdf">Correspondence, 1974-1975</a>

%H B. Poonen and M. Rubinstein (1998) <a href="http://math.mit.edu/~poonen/papers/ngon.pdf">The Number of Intersection Points Made by the Diagonals of a Regular Polygon</a>, SIAM J. Discrete Mathematics 11(1), pp. 135-156, doi:<a href="http://dx.doi.org/10.1137/S0895480195281246">10.1137/S0895480195281246</a>, arXiv:<a href="http://arXiv.org/abs/math.MG/9508209">math.MG/9508209</a> (has fewer typos than the SIAM version)

%H B. Poonen and M. Rubinstein, <a href="http://math.mit.edu/~poonen/papers/ngon.m">Mathematica programs for these sequences</a>

%H M. Rubinstein, <a href="/A006561/a006561_3.pdf">Drawings for n=4,5,6,...</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CircuitRank.html">Circuit Rank</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolygonDiagonalIntersectionGraph.html">Polygon Diagonal Intersection Graph</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RegularPolygonDivisionbyDiagonals.html">Regular Polygon Division by Diagonals</a>

%H <a href="/index/Pol#Poonen">Sequences formed by drawing all diagonals in regular polygon</a>

%H <a href="/index/Ch#CHORD">Sequences related to chord diagrams</a>

%F 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

%F For odd n, a(n) = (24 - 42n + 23n^2 - 6n^3 + n^4)/24. - _Graeme McRae_, Dec 24 2004

%F a(n) = A006533(n) - n. - T. D. Noe, Dec 23 2006

%F For odd n, binomial transform of [1, 10, 29, 36, 16, 0, 0, 0,...] = [1, 11, 50, 154,...]. - Gary W. Adamson, Aug 02 2011

%F a(n) = A135565(n) - A007569(n) + 1. [From Max Alekseyev]

%t 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

%o (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; }

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

%K easy,nonn,nice

%O 1,4

%A _N. J. A. Sloane_, Bjorn Poonen (poonen(AT)math.princeton.edu)

%E More terms from _Graeme McRae_, Dec 26 2004

%E a(1)=a(2)=0 prepended by _Max Alekseyev_, Dec 01 2011

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Last modified November 20 18:55 EST 2018. Contains 317413 sequences. (Running on oeis4.)