A293846 Numbers such that k is the altitude of a Heronian triangle with sides m-13, m, m+13.

9, 24, 39, 60, 105, 156, 231, 396, 585, 864, 1479, 2184, 3225, 5520, 8151, 12036, 20601, 30420, 44919, 76884, 113529, 167640, 286935, 423696, 625641, 1070856, 1581255, 2334924, 3996489, 5901324, 8714055, 14915100, 22024041, 32521296, 55663911, 82194840

Offset

0,1

Comments

a(n) gives the values of y satifacting 3*x^2 - y^2 = 507; corresponding x values are given by A293817.

a(n)/3 is the radius of the inscribed circle.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,4,0,0,-1).

Formula

a(n) = 4*a(n-3) - a(n-6), a(1)=9, a(2)=24, a(3)=39, a(4)=60, a(5)=105, a(6)=156.

G.f.: 3*(3 + 8*x + 13*x^2 + 8*x^3 + 3*x^4) / (1 - 4*x^3 + x^6). - Colin Barker, Dec 27 2017

Example

If the sides are 15, 28, 41 the triangle has the altitude 9 and is a part of the Pythagorean triangle with the sides 9, 40, 41, so 9 is a term.

Mathematica

CoefficientList[ Series[ 3(3x^4 +8x^3 +13x^2 +8x +3)/(x^6 -4x^3 +1), {x, 0, 35}], x] (* or *)
LinearRecurrence[{0, 0, 4, 0, 0, -1}, 3 {3, 8, 13, 20, 35, 52}, 36] (* Robert G. Wilson v, Dec 27 2017 *)

Prog

(PARI) Vec(3*(3 + 8*x + 13*x^2 + 8*x^3 + 3*x^4) / (1 - 4*x^3 + x^6) + O(x^40)) \\ Colin Barker, Dec 27 2017

Crossrefs

Cf. A003500, A005320, A296795, A296796.

Sequence in context: A345843 A047720 A320599 * A212462 A161449 A063066

Adjacent sequences: A293843 A293844 A293845 * A293847 A293848 A293849

Keyword

nonn,easy

Author

Sture Sjöstedt, Dec 27 2017

Status

approved