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%I #51 Dec 12 2022 03:15:25
%S 0,6,84,1170,16296,226974,3161340,44031786,613283664,8541939510,
%T 118973869476,1657092233154,23080317394680,321467351292366,
%U 4477462600698444,62363009058485850,868604664218103456,12098102289994962534,168504827395711372020,2346969481249964245746
%N Areas of almost-equilateral Heronian triangles (integral side lengths m-1, m, m+1 and integral area).
%C Corresponding m's are in A016064. Corresponding values of lesser side give A016064.
%H Vincenzo Librandi, <a href="/A011945/b011945.txt">Table of n, a(n) for n = 1..890</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H E. Keith Lloyd, <a href="http://www.jstor.org/stable/3619201">The Standard Deviation of 1, 2,..., n: Pell's Equation and Rational Triangles</a>, Math. Gaz. vol 81 (1997), 231-243.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Heronian_triangle">Heronian triangle</a>
%H P. Yiu, <a href="http://math.fau.edu/yiu/RecreationalMathematics2003.pdf">Heron triangles with consecutive sides</a>, Recreational Mathematics, Chap. 9.3, pp. 80/360. (This is a download of 360 pages.)
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14,-1).
%F s(n) = floor((a+1)/4)*sqrt(3*(a+3)*(a-1)), where a = A016064(n). - _Zak Seidov_, Feb 23 2005
%F a(n) = 14*a(n-1) - a(n-2); a(1) = 0, a(2) = 6.
%F G.f.: 6*x^2/(1 - 14*x + x^2). - _Philippe Deléham_, Nov 17 2008
%F a(n) = (s/4)*((7 + 4*s)^n - (7 - 4*s)^n), where s = sqrt(3). - _Zak Seidov_, Apr 02 2014
%F 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
%t CoefficientList[Series[6 x/(1 - 14 x + x^2), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 15 2013 *)
%t LinearRecurrence[{14,-1},{0,6},20] (* _Harvey P. Dale_, Jan 24 2015 *)
%Y Equals 6 * A007655(n+1).
%Y Cf. this sequence (areas), A334277 (perimeters).
%Y Cf. A003500 (middle side lengths), A016064 (smallest side lengths), A335025 (largest side lengths).
%Y Cf. A102341, A103974, A103975.
%K nonn,easy
%O 1,2
%A E. K. Lloyd
%E Entry revised by _N. J. A. Sloane_, Feb 03 2007
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