|
%I #35 Jul 31 2022 19:32:20
%S -3,-2,0,3,7,12,18,25,33,42,52,63,75,88,102,117,133,150,168,187,207,
%T 228,250,273,297,322,348,375,403,432,462,493,525,558,592,627,663,700,
%U 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
%N Difference between the number of diagonals and the number of sides for a convex n-gon.
%C The number of diagonals for a convex polygon with n sides is n*(n-3)/2.
%C 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.
%H G. C. Greubel, <a href="/A307681/b307681.txt">Table of n, a(n) for n = 3..1000</a>
%H Ask Dr. Math, <a href="http://mathforum.org/library/drmath/sets/select/dm_polygon_diagonals.html">Polygon diagonals</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polygon.html">Polygon</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PolygonDiagonal.html">Polygon diagonal</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = n*(n - 5)/2.
%F a(n) = binomial(n-2, 2) - 3. - _Yuchun Ji_, Aug 12 2021
%F From _G. C. Greubel_, Jul 31 2022: (Start)
%F G.f.: (-1)*x^3*(3 - 7*x + 3*x^2)/(1-x)^3.
%F E.g.f.: (x/2)*( (x-4)*exp(x) + 4 + 3*x ). (End)
%t Table[(n(n-5))/2, {n,3,80}] (* _Harvey P. Dale_, Jan 23 2021 *)
%o (Magma) [n*(n-5)/2: n in [3..80]]; // _G. C. Greubel_, Jul 31 2022
%o (Sage) [n*(n-5)/2 for n in (3..80)] # _G. C. Greubel_, Jul 31 2022
%Y Cf. A000096 (number of diagonals of an n-gon).
%Y Cf. A006561 (number of intersections of diagonals in the interior of regular n-gon).
%Y Cf. A007678 (number of regions in regular n-gon with all diagonals drawn).
%K sign,easy
%O 3,1
%A _Bernard Schott_, Apr 21 2019
|
