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 A007678 Number of regions in regular n-gon with all diagonals drawn. (Formerly M3411) 143
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 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 number of 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 Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe) Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918. Robert Dougherty-Bliss, First draft of Python program to produce colored drawings of these figures, Github, Feb 09 2020. M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5. M. F. Hasler, Interactive illustration of A006561(n) & A006533(n); colored version for n=6 and for n=8. Sascha Kurz, m-gons in regular n-gons (in German). J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975 B. Poonen and M. Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11, no. 1 (1998), pp. 135-156; DOI:10.1137/S0895480195281246. [Author's copy]. The latest arXiv version arXiv:math/9508209 has corrected some typos in the published version. B. Poonen and M. Rubinstein, Mathematica programs for these sequences J. F. Rigby, Multiple intersections of diagonals of regular polygons, and related topics, Geom. Dedicata 9 (1980), 207-238. M. Rubinstein, Drawings for n=4,5,6,... Scott R. Shannon, Colored illustration for n = 17 Scott R. Shannon, Colored illustration for n = 18 Scott R. Shannon, Colored illustration for n = 19 Scott R. Shannon, Colored illustration for n = 23 Scott R. Shannon, Colored illustration for n = 27 Scott R. Shannon, Colored illustration for n = 40 Scott R. Shannon, Colored illustration for n = 41 (1st version) Scott R. Shannon, Colored illustration for n = 41 (2nd version) Scott R. Shannon, Colored illustration for n = 41 (3rd version). This variation has coloring based on the number of edges of the polygon: red = 3-gon, orange = 4-gon, yellow = 5-gon, light-green = 6-gon etc. N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence. N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 18. 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 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)*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 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) /* 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 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. Sequence in context: A260057 A260150 A258472 * A339493 A364282 A159350 Adjacent sequences: A007675 A007676 A007677 * A007679 A007680 A007681 KEYWORD 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

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Last modified September 14 13:32 EDT 2024. Contains 375921 sequences. (Running on oeis4.)