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

Author

Juri-Stepan Gerasimov, Jul 13 2009

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A001075, A003500, A011945, A016064.

Programs

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

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 *)

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

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

Extensions

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