

A007678


Number of regions in regular ngon 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
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OFFSET

1,4


COMMENTS

This sequence and A006533 are two equivalent ways of presenting the same sequence.
Also the circuit rank of the npolygon 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 ngon. 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 nsimplex 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 nsimplex.  Mohammed Yaseen, Nov 05 2022


REFERENCES

Jean Meeus, Wiskunde Post (Belgium), Vol. 10, 1972, pp. 6263.
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

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.


FORMULA

For odd n > 3, a(n) = sumstep {i=5, n, 2, (i2)*floor(n/2)+(i4)*ceiling(n/2)+1} + x*(x+1)*(2*x+1)/6*n), where x = (n5)/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
For odd n, binomial transform of [1, 10, 29, 36, 16, 0, 0, 0, ...] = [1, 11, 50, 154, ...].  Gary W. Adamson, Aug 02 2011
See the Mma code in A006533 for the explicit PoonenRubenstein 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^46n^3+23n^242n+24)/24 + del[2, n](5n^3+42n^240n48)/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=(fn1)^21; nr=fn2*n+fn+(n2)*fn+cn; x=(n5)/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, (((n6)*n+23)*n42)*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

A187781 gives number of distinct regions.


KEYWORD

nonn,nice


AUTHOR



EXTENSIONS



STATUS

approved



