A228517 Area of the Robbins pentagons.

276, 342, 1332, 1638, 1848, 1884, 2058, 2094, 2148, 2268, 2358, 2424, 2436, 2760, 2844, 2856, 2952, 3108, 3150, 3276, 3390, 3624, 3696, 3810, 4044, 4056, 3624, 4512, 4998, 5112, 5268, 6192, 6312, 6786, 6942, 6984, 7230, 7374, 7476, 7962, 7680, 7722, 8022, 8712

Offset

1,1

Comments

See the first link and the table 2 page 23 for the results of the exploration of all pentagons with perimeter less than 400.

A Robbins pentagon is a cyclic polygon with 5 integer sides and integer area.

Any Robbins pentagon with five integer sides has integer area (proof in reference).

Theorem (Robbins). Consider a cyclic pentagon with sides a,b,c,d,e and area A. If s1, s2, s3, s4 and s5 are the symmetric polynomials in the squares of the sides, x = 16A^2, t=x-4*s2 + s1^2, u = 8*s3 + s1*t2, v = -64*s4 + t^2 and w = 128*s5, then u (hence the square of the area) satisfies the condition: x*v^3 + u^2 * v^2 - 18*x*u*v*w - 27*x^2*w^2.

Links

Table of n, a(n) for n=1..44.

Ralph H. Buchholz and James A. MacDougall, Cyclic polygons with rational Sides and Area, Journal of Number Theory, Volume 128, Issue 1, January 2008, Pages 17-48.

Kival Ngaokrajang, Illustration for n = 1..6

D. P. Robbins, Areas of polygons inscribed in a circle, Amer. Math. Monthly, 102 (1995), 523-530.

Eric Weisstein's World of Mathematics, Cyclic Pentagon

Example

276 is the area of the pentagon with sides (7, 7, 15, 15, 24).

Crossrefs

Sequence in context: A003903 A131884 A284279 * A008892 A216072 A284277

Adjacent sequences: A228514 A228515 A228516 * A228518 A228519 A228520

Keyword

nonn

Author

Michel Lagneau, Aug 24 2013

Status

approved