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A011945
Areas of almost-equilateral Heronian triangles (integral side lengths m-1, m, m+1 and integral area).
14
0, 6, 84, 1170, 16296, 226974, 3161340, 44031786, 613283664, 8541939510, 118973869476, 1657092233154, 23080317394680, 321467351292366, 4477462600698444, 62363009058485850, 868604664218103456, 12098102289994962534, 168504827395711372020, 2346969481249964245746
OFFSET
1,2
COMMENTS
Corresponding m's are in A016064. Corresponding values of lesser side give A016064.
LINKS
Tanya Khovanova, Recursive Sequences
E. Keith Lloyd, The Standard Deviation of 1, 2,..., n: Pell's Equation and Rational Triangles, Math. Gaz. vol 81 (1997), 231-243.
Eric Weisstein's World of Mathematics, Heronian Triangle
P. Yiu, Heron triangles with consecutive sides, Recreational Mathematics, Chap. 9.3, pp. 80/360. (This is a download of 360 pages.)
FORMULA
s(n) = floor((a+1)/4)*sqrt(3*(a+3)*(a-1)), where a = A016064(n). - Zak Seidov, Feb 23 2005
a(n) = 14*a(n-1) - a(n-2); a(1) = 0, a(2) = 6.
G.f.: 6*x^2/(1 - 14*x + x^2). - Philippe Deléham, Nov 17 2008
a(n) = (s/4)*((7 + 4*s)^n - (7 - 4*s)^n), where s = sqrt(3). - Zak Seidov, Apr 02 2014
E.g.f.: 6 - exp(7*x)*(12*cosh(4*sqrt(3)*x) - 7*sqrt(3)*sinh(4*sqrt(3)*x))/2. - Stefano Spezia, Dec 12 2022
MATHEMATICA
CoefficientList[Series[6 x/(1 - 14 x + x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 15 2013 *)
LinearRecurrence[{14, -1}, {0, 6}, 20] (* Harvey P. Dale, Jan 24 2015 *)
CROSSREFS
Equals 6 * A007655(n+1).
Cf. this sequence (areas), A334277 (perimeters).
Cf. A003500 (middle side lengths), A016064 (smallest side lengths), A335025 (largest side lengths).
Sequence in context: A248338 A144514 A244284 * A113888 A163947 A306244
KEYWORD
nonn,easy
AUTHOR
E. K. Lloyd
EXTENSIONS
Entry revised by N. J. A. Sloane, Feb 03 2007
STATUS
approved