A211379 -id:A211379 - cp's OEIS frontend

A367204 Number of diagonals in a regular n-gon that are parallel to an edge.

0, 0, 5, 3, 14, 8, 27, 15, 44, 24, 65, 35, 90, 48, 119, 63, 152, 80, 189, 99, 230, 120, 275, 143, 324, 168, 377, 195, 434, 224, 495, 255, 560, 288, 629, 323, 702, 360, 779, 399, 860, 440, 945, 483, 1034, 528, 1127, 575, 1224, 624, 1325, 675, 1430, 728, 1539, 783

Offset: 3

Paolo Xausa, Nov 10 2023

A diagonal is parallel to an edge if and only if, on at least one side of the diagonal, there is an odd number of edges.

If n is odd, all of the diagonals of the n-gon are parallel to an edge.

Paolo Xausa, Table of n, a(n) for n = 3..10000
Thomas Andrews, Number of Parallel/Not Parallel Diagonals of a Regular Polygon, Mathematics Stack Exchange, 2014.
Paolo Xausa, Illustration of first terms.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Even-indexed terms of A000096 interleaved with A005563.

Cf. A002378, A211379.

LinearRecurrence[{0,3,0,-3,0,1},{0,0,5,3,14,8},100] (* or *)
A367204[n_]:=If[OddQ[n],n(n-3)/2,n(n-4)/4];Array[A367204,100,3]

def A367204(n): return n*(n-3)>>1 if n&1 else n*(n-4)>>2 # Chai Wah Wu, Nov 22 2023

a(n) = n(n-3)/2 = A000096(n-3) if n is odd;
a(n) = n(n-4)/4 = A005563((n-4)/2) = A000096(n-3) - A002378(n/2-1) if n is even.
Sum_{n>=5} 1/a(n) = 59/36 - 2*log(2)/3. - Amiram Eldar, Dec 06 2024

A211380 Number of pairs of intersecting diagonals in the interior and exterior of a regular n-gon.

Original entry on oeis.org

0, 1, 5, 15, 42, 94, 189, 340, 572, 903, 1365, 1981, 2790, 3820, 5117, 6714, 8664, 11005, 13797, 17083, 20930, 25386, 30525, 36400, 43092, 50659, 59189, 68745, 79422, 91288, 104445, 118966, 134960, 152505, 171717, 192679, 215514, 240310, 267197, 296268, 327660

Offset: 3

Martin Renner, Feb 07 2013

Eric Weisstein, Regular Polygon Division by Diagonals (MathWorld).
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A176145, A211379.

a:=n->piecewise(n mod 2 = 0,1/8*n*(n^3-11*n^2+43*n-58),n mod 2 = 1,1/8*n*(n-3)*(n^2-8*n+19),0);

Drop[CoefficientList[Series[x^4(2x^5-3x^4-7x^3-x^2-2x-1)/((x-1)^5(x+1)^2),{x,0,50}],x],3] (* or *) LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,1,5,15,42,94,189},50] (* Harvey P. Dale, Dec 03 2022 *)

def A211380(n): return n*(n*(n*(n-11)+43)-58+(n&1))>>3 # Chai Wah Wu, Nov 22 2023

a(n) = 1/8*n*(n^3-11*n^2+43*n-58) for n even;
a(n) = 1/8*n*(n-3)*(n^2-8*n+19) for n odd.
a(n) = A176145(n) - A211379(n).
G.f.: x^4*(2*x^5-3*x^4-7*x^3-x^2-2*x-1) / ((x-1)^5*(x+1)^2). [Colin Barker, Feb 14 2013]

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A367204 Number of diagonals in a regular n-gon that are parallel to an edge.

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