%I #33 Oct 10 2023 10:18:05

%S 13,14,19,26,37,62,91,134,229,338,499,854,1261,1862,3187,4706,6949,

%T 11894,17563,25934,44389,65546,96787,165662,244621,361214,618259,

%U 912938,1348069,2307374,3407131,5031062,8611237,12715586,18776179,32137574,47455213,70073654

%N Numbers k such that m=2*k is the middle side in a Heronian triangle with sides m-13, m , m+13.

%C a(n) gives values of x satisfying 3*x^2 - y^2 = 507; corresponding y values are given by A293846.

%H Colin Barker, <a href="/A293817/b293817.txt">Table of n, a(n) for n = 0..1000</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 a(n) = 4*a(n-3)-a(n-6), a(1)= 13, a(2)= 14, a(3)= 19, a(4)= 26, a(5)= 37, a(6)= 62.

%F G.f.: (13 + 14*x + 19*x^2 - 26*x^3 - 19*x^4 - 14*x^5) / (1 - 4*x^3 + x^6). - _Colin Barker_, Dec 27 2017

%e The smallest triangle of this type with 3 acute angles has the sides: 61, 74, 87.

%t LinearRecurrence[{0,0,4,0,0,-1},{13,14,19,26,37,62},40] (* _Harvey P. Dale_, Oct 10 2023 *)

%o (PARI) Vec((13 + 14*x + 19*x^2 - 26*x^3 - 19*x^4 - 14*x^5) / (1 - 4*x^3 + x^6) + O(x^40)) \\ _Colin Barker_, Dec 27 2017

%Y Cf. A003500, A005320, A296795, A296796.

%K nonn,easy

%O 0,1

%A _Sture Sjöstedt_, Dec 27 2017