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A006561 Number of intersections of diagonals in the interior of regular n-gon.
(Formerly M3833)
31
0, 0, 0, 1, 5, 13, 35, 49, 126, 161, 330, 301, 715, 757, 1365, 1377, 2380, 1837, 3876, 3841, 5985, 5941, 8855, 7297, 12650, 12481, 17550, 17249, 23751, 16801, 31465, 30913, 40920, 40257, 52360, 46981, 66045, 64981, 82251, 80881, 101270 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

REFERENCES

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

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 1000 terms from T. D. Noe]

Jessica Gonzalez, Illustration of a(4) through a(9)

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), Sep 01 2017. (For colored versions see A006533.)

Sascha Kurz, m-gons in regular n-gons

B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006, which has fewer typos than the SIAM version.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156 (1998). [Copy on SIAM web site]

B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998). [Copy on B. Poonen's web site]

B. Poonen and M. Rubinstein, Mathematica programs for A006561 and related sequences

M. Rubinstein, Drawings for n=4,5,6,...

N. J. A. Sloane, Illustrations of a(8) and a(9)

N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)

R. G. Wilson V, Illustration of a(10)

Index entry for Sequences formed by drawing all diagonals in regular polygon

FORMULA

Let delta(m,n) = 1 if m divides n, otherwise 0.

For n >= 3, a(n) = binomial(4,4) + (-5*n^3 + 45*n^2 - 70*n + 24)*delta(2,n)/24

- (3*n/2)*delta(4,n) + (-45*n^2 + 262*n)*delta(6,n)/6 + 42*n*delta(12,n)

+ 60*n*delta(18,n) + 35*n*delta(24,n) - 38*n*delta(30,n)

- 82*n*delta(42,n) - 330*n*delta(60,n) - 144*n*delta(84,n)

- 96*n*delta(90,n) - 144*n*delta(120,n) - 96*n*delta(210,n). [Poonen and Rubinstein, Theorem 1] - N. J. A. Sloane, Aug 09 2017

For odd n, binomial(n,4) = (n^4-6n^3+11n^2-6n)/24. For even n, use this formula, but then subtract 2 for every 3-crossing, subtract 5 for every 4-crossing, subtract 9 for every 5-crossing, etc. The number to be subtracted is one smaller than a triangular number. - Graeme McRae, Dec 26 2004

a(n) = A007569(n)-n. - T. D. Noe, Dec 23 2006

a(2n+5) = A053126(n+4). - Philippe Deléham, Jun 07 2013

MAPLE

delta:=(m, n) -> if (n mod m) = 0 then 1 else 0; fi;

f:=proc(n) global delta;

if n <= 2 then 0 else \

binomial(n, 4)  \

+ (-5*n^3 + 45*n^2 - 70*n + 24)*delta(2, n)/24 \

- (3*n/2)*delta(4, n) \

+ (-45*n^2 + 262*n)*delta(6, n)/6  \

+ 42*n*delta(12, n) \

+ 60*n*delta(18, n) \

+ 35*n*delta(24, n) \

- 38*n*delta(30, n) \

- 82*n*delta(42, n) \

- 330*n*delta(60, n) \

- 144*n*delta(84, n) \

- 96*n*delta(90, n) \

- 144*n*delta(120, n) \

- 96*n*delta(210, n); fi; end;

[seq(f(n), n=1..100)]; # N. J. A. Sloane, Aug 09 2017

MATHEMATICA

del[m_, n_]:=If[Mod[n, m]==0, 1, 0]; Int[n_]:=If[n<4, 0, Binomial[n, 4] + del[2, n](-5n^3+45n^2-70n+24)/24 - del[4, n](3n/2) + del[6, n](-45n^2+262n)/6 + del[12, n]*42n + del[18, n]*60n + del[24, n]*35n - del[30, n]*38n - del[42, n]*82n - del[60, n]*330n - del[84, n]*144n - del[90, n]*96n - del[120, n]*144n - del[210, n]*96n]; Table[Int[n], {n, 1, 1000}] (* T. D. Noe, Dec 21 2006 *)

PROG

(PARI) {a(n, m=[4, 6, 12, 18, 24, 30, 42, 60, 84, 90, 120, 210], v=[-3/2, (262-45*n)/6, 42, 60, 35, -38, -82, -330, -144, -96, -144, -96])=if(n>3, binomial(n, 4) + if(n%2==0, (-5*n^2+45*n-70)*n/24+1 + sum(i=1, #m, if(n%m[i]==0, v[i]))*n))} \\ M. F. Hasler, Aug 23 2017

CROSSREFS

Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.

See also A101363, A292104, A292105.

See A290447 for an analogous problem on a line.

Sequence in context: A034521 A092647 A171262 * A146845 A192310 A167710

Adjacent sequences:  A006558 A006559 A006560 * A006562 A006563 A006564

KEYWORD

easy,nonn,nice,changed

AUTHOR

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

STATUS

approved

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Last modified October 18 00:56 EDT 2017. Contains 293484 sequences.