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A211379 Number of pairs of parallel diagonals in a regular n-gon. 1
0, 0, 0, 3, 7, 16, 27, 45, 66, 96, 130, 175, 225, 288, 357, 441, 532, 640, 756, 891, 1035, 1200, 1375, 1573, 1782, 2016, 2262, 2535, 2821, 3136, 3465, 3825, 4200, 4608, 5032, 5491, 5967, 6480, 7011, 7581, 8170, 8800, 9450, 10143, 10857, 11616, 12397, 13225 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,4

LINKS

Table of n, a(n) for n=3..50.

Eric Weisstein, Regular Polygon Division by Diagonals (MathWorld).

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

FORMULA

a(n) = 1/2*n*(binomial(1/2*(n-2),2)+binomial(1/2*(n-4),2)) = 1/8*n*(n-4)^2 for n even;

a(n) = n*binomial(1/2*(n-3),2) = 1/8*n*(n-3)*(n-5) for n odd.

G.f.: -x^6*(x^2-x-3) / ((x-1)^4*(x+1)^2). [Colin Barker, Feb 14 2013]

EXAMPLE

a(6) = 3 since by numbering the vertices from 1 to 6 there are three pairs of parallel diagonals, i. e. {[1, 3], [4, 6]}, {[1, 5], [2, 4]}, {[2, 6], [3, 5]}.

a(7) = 7 since there are the seven pairs {[1, 3], [4, 7]}, {[1, 4], [5, 7]}, {[1, 5], [2, 4]}, {[1, 6], [2, 5]}, {[2, 6], [3, 5]}, {[2, 7], [3, 6]}, {[3, 7], [4, 6]}.

a(8) = 16 since there are the sixteen pairs {[1, 3], [4, 8]}, {[1, 3], [5, 7]}, {[1, 4], [5, 8]}, {[1, 5], [2, 4]}, {[1, 5], [6, 8]}, {[1, 6], [2, 5]}, {[1, 7], [2, 6]}, {[1, 7], [3, 5]}, {[2, 4], [6, 8]}, {[2, 6], [3, 5]}, {[2, 7], [3, 6]}, {[2, 8], [3, 7]}, {[2, 8], [4, 6]}, {[3, 7], [4, 6]}, {[3, 8], [4, 7]}, {[4, 8], [5, 7]}.

MAPLE

a:=n->piecewise(n mod 2 = 0, 1/8*n*(n-4)^2, n mod 2 = 1, 1/8*n*(n-3)*(n-5), 0);

CROSSREFS

Cf. A000096.

Sequence in context: A036666 A218359 A117491 * A213180 A110585 A184677

Adjacent sequences:  A211376 A211377 A211378 * A211380 A211381 A211382

KEYWORD

nonn,easy

AUTHOR

Martin Renner, Feb 07 2013

STATUS

approved

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Last modified December 12 01:57 EST 2019. Contains 329948 sequences. (Running on oeis4.)