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 A228517 Area of the Robbins pentagons. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 16 20:53 EDT 2024. Contains 375977 sequences. (Running on oeis4.)