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A006561 Number of intersections of diagonals in the interior of regular n-gon.
(Formerly M3833)
27
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

T. D. Noe, Table of n, a(n) for n=1..1000

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

Sascha Kurz, m-gons in regular n-gons

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.

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).

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, Mathematica programs for these sequences

B. Poonen & M. Rubinstein, The Number Of Intersection Points Made By The Diagonals Of A Regular Polygon, SIAM Journal in Discrete Mathematics, pp. 135-6 vol. 11 no.1 1998.

Sequences formed by drawing all diagonals in regular polygon

FORMULA

For odd n, (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

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 *)

CROSSREFS

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

Sequence in context: A034521 A092647 A171262 * A146845 A192310 A167710

Adjacent sequences:  A006558 A006559 A006560 * A006562 A006563 A006564

KEYWORD

easy,nonn,nice

AUTHOR

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

STATUS

approved

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Last modified June 24 05:50 EDT 2017. Contains 288697 sequences.