A151961 Semiperimeter of the n-th Heronian triangle.

3, 6, 21, 78, 291, 1086, 4053, 15126, 56451, 210678, 786261, 2934366, 10951203, 40870446, 152530581, 569251878, 2124476931, 7928655846, 29590146453, 110431929966, 412137573411, 1538118363678, 5740335881301, 21423225161526, 79952564764803, 298387033897686

Offset

1,1

Comments

The side lengths are consecutive integers (A016064) and the area is an integer (A011945).

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

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

G. Jacob Martens, Rational right triangles and the Congruent Number Problem, arXiv:2112.09553 [math.GM], 2021, see section 10.1 The Brahmaguptra triangles, equation (99).

Index entries for linear recurrences with constant coefficients, signature (4,-1).

Formula

a(n) = 3 * A001075(n-1). - Joerg Arndt, Oct 10 2022

a(n) = 3*(A016064(n-1) + 1)/2 = 3*A003500(n-1)/2. - R. J. Mathar, Jul 27 2009

From Colin Barker, Mar 30 2012: (Start)

a(n) = 4*a(n-1) - a(n-2).

G.f.: 3*x*(1-2*x)/(1-4*x+x^2). (End)

a(n) = 3*( (2+sqrt(3))*(2-sqrt(3))^n + (2-sqrt(3))*(2+sqrt(3))^n )/2. - Colin Barker, Oct 12 2015

Mathematica

CoefficientList[Series[3(1-2x)/(1-4x+x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 11 2014 *)
3*ChebyshevT[Range[0, 40], 2] (* G. C. Greubel, Oct 10 2022 *)
LinearRecurrence[{4, -1}, {3, 6}, 30] (* Harvey P. Dale, Dec 21 2022 *)

Prog

(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
(PARI) Vec(3*x*(1-2*x)/(1-4*x+x^2) + O(x^40)) \\ Colin Barker, Oct 12 2015
(SageMath) [3*chebyshev_T(n, 2) for n in range(41)] # G. C. Greubel, Oct 10 2022

Crossrefs

Cf. A001075, A003500, A011945, A016064.

Sequence in context: A261582 A135686 A218244 * A025229 A073951 A032098

Adjacent sequences: A151958 A151959 A151960 * A151962 A151963 A151964

Keyword

nonn,easy

Author

Juri-Stepan Gerasimov, Jul 13 2009

Extensions

More terms from R. J. Mathar, Jul 27 2009

Status

approved