A307681 Difference between the number of diagonals and the number of sides for a convex n-gon.

-3, -2, 0, 3, 7, 12, 18, 25, 33, 42, 52, 63, 75, 88, 102, 117, 133, 150, 168, 187, 207, 228, 250, 273, 297, 322, 348, 375, 403, 432, 462, 493, 525, 558, 592, 627, 663, 700, 738, 777, 817, 858, 900, 943, 987, 1032, 1078, 1125, 1173, 1222, 1272, 1323, 1375, 1428, 1482, 1537, 1593, 1650, 1708, 1767, 1827, 1888, 1950, 2013, 2077, 2142, 2208, 2275

Offset

3,1

Comments

The number of diagonals for a convex polygon with n sides is n*(n-3)/2.

For a triangle and a quadrilateral, the number of sides is greater than the number of diagonals. For a pentagon, the number of sides is equal to the number of diagonals. For an hexagon or a polygon with more than six sides, the number of diagonals is greater than the number of sides.

Links

G. C. Greubel, Table of n, a(n) for n = 3..1000

Ask Dr. Math, Polygon diagonals. [Wayback Machine link]

Eric Weisstein's World of Mathematics, Polygon.

Eric Weisstein's World of Mathematics, Polygon diagonal.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

a(n) = n*(n - 5)/2.

a(n) = binomial(n-2, 2) - 3. - Yuchun Ji, Aug 12 2021

From G. C. Greubel, Jul 31 2022: (Start)

G.f.: (-1)*x^3*(3 - 7*x + 3*x^2)/(1-x)^3.

E.g.f.: (x/2)*( (x-4)*exp(x) + 4 + 3*x ). (End)

From Amiram Eldar, Nov 01 2025: (Start)

Sum_{n>=6} 1/a(n) = 137/150.

Sum_{n>=6} (-1)^n/a(n) = 4*log(2)/5 - 47/150. (End)

Mathematica

Table[(n(n-5))/2, {n, 3, 80}] (* Harvey P. Dale, Jan 23 2021 *)

Prog

(Magma) [n*(n-5)/2: n in [3..80]]; // G. C. Greubel, Jul 31 2022
(SageMath) [n*(n-5)/2 for n in (3..80)] # G. C. Greubel, Jul 31 2022

Crossrefs

Cf. A000096 (number of diagonals of an n-gon).

Cf. A006561 (number of intersections of diagonals in the interior of regular n-gon).

Cf. A007678 (number of regions in regular n-gon with all diagonals drawn).

Sequence in context: A231132 A290327 A131732 * A331922 A198826 A088161

Adjacent sequences: A307678 A307679 A307680 * A307682 A307683 A307684

Keyword

sign,easy

Author

Bernard Schott, Apr 21 2019

Status

approved