%I #32 Dec 28 2017 10:34:21

%S 13,14,22,38,43,77,139,158,286,518,589,1067,1933,2198,3982,7214,8203,

%T 14861,26923,30614,55462,100478,114253,206987,374989,426398,772486,

%U 1399478,1591339,2882957,5222923,5938958,10759342,19492214,22164493,40154411,72745933

%N Numbers k such that m = 2*k is the middle side in a Heronian triangle with sides m - 11, m, m + 11.

%C a(n) gives values of x satisfying 3*x^2 - y^2 = 363; the corresponding y values are given by A296796.

%H Colin Barker, <a href="/A296795/b296795.txt">Table of n, a(n) for n = 0..1000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Heronian_triangle">Heronian triangle</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,4,0,0,-1).

%F From _Colin Barker_, Dec 22 2017: (Start)

%F G.f.: (13 + 14*x + 22*x^2 - 14*x^3 - 13*x^4 - 11*x^5) / (1 - 4*x^3 + x^6).

%F a(n) = 4*a(n-3) - a(n-6) for n>5.

%F (End)

%e The smallest triangle of this type has following sides: 15, 26, 37 has the altitude 12 and is a part of the Pythagorean triangle with sides : 12, 35, 37.

%t CoefficientList[Series[(13 + 14 x + 22 x^2 - 14 x^3 - 13 x^4 - 11 x^5)/(1 - 4 x^3 + x^6), {x, 0, 36}], x] (* _Michael De Vlieger_, Dec 22 2017 *)

%o (PARI) Vec((13 + 14*x + 22*x^2 - 14*x^3 - 13*x^4 - 11*x^5) / (1 - 4*x^3 + x^6) + O(x^40)) \\ _Colin Barker_, Dec 22 2017

%Y Cf. A003500, A005320, A296796.

%K nonn,easy

%O 0,1

%A _Sture Sjöstedt_, Dec 20 2017

%E More terms from _Colin Barker_, Dec 22 2017