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%I #44 Dec 21 2022 17:56:49
%S 3,6,21,78,291,1086,4053,15126,56451,210678,786261,2934366,10951203,
%T 40870446,152530581,569251878,2124476931,7928655846,29590146453,
%U 110431929966,412137573411,1538118363678,5740335881301,21423225161526,79952564764803,298387033897686
%N Semiperimeter of the n-th Heronian triangle.
%C The side lengths are consecutive integers (A016064) and the area is an integer (A011945).
%C Except for the first term, positive values of x (or y) satisfying x^2 - 4*x*y + y^2 + 27 = 0. - _Colin Barker_, Feb 08 2014
%C Except for the first term, positive values of x (or y) satisfying x^2 - 14*x*y + y^2 + 432 = 0. - _Colin Barker_, Feb 16 2014
%H Vincenzo Librandi, <a href="/A151961/b151961.txt">Table of n, a(n) for n = 1..200</a>
%H G. Jacob Martens, <a href="https://arxiv.org/abs/2112.09553">Rational right triangles and the Congruent Number Problem</a>, arXiv:2112.09553 [math.GM], 2021, see section 10.1 The Brahmaguptra triangles, equation (99).
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1).
%F a(n) = 3 * A001075(n-1). - _Joerg Arndt_, Oct 10 2022
%F a(n) = 3*(A016064(n-1) + 1)/2 = 3*A003500(n-1)/2. - _R. J. Mathar_, Jul 27 2009
%F From _Colin Barker_, Mar 30 2012: (Start)
%F a(n) = 4*a(n-1) - a(n-2).
%F G.f.: 3*x*(1-2*x)/(1-4*x+x^2). (End)
%F a(n) = 3*( (2+sqrt(3))*(2-sqrt(3))^n + (2-sqrt(3))*(2+sqrt(3))^n )/2. - _Colin Barker_, Oct 12 2015
%t CoefficientList[Series[3(1-2x)/(1-4x+x^2), {x,0,30}], x] (* _Vincenzo Librandi_, Feb 11 2014 *)
%t 3*ChebyshevT[Range[0, 40], 2] (* _G. C. Greubel_, Oct 10 2022 *)
%t LinearRecurrence[{4,-1},{3,6},30] (* _Harvey P. Dale_, Dec 21 2022 *)
%o (Magma) I:=[3,6]; [n le 2 select I[n] else 4*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Feb 11 2014
%o (PARI) Vec(3*x*(1-2*x)/(1-4*x+x^2) + O(x^40)) \\ _Colin Barker_, Oct 12 2015
%o (SageMath) [3*chebyshev_T(n, 2) for n in range(41)] # _G. C. Greubel_, Oct 10 2022
%Y Cf. A001075, A003500, A011945, A016064.
%K nonn,easy
%O 1,1
%A _Juri-Stepan Gerasimov_, Jul 13 2009
%E More terms from _R. J. Mathar_, Jul 27 2009
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