%I #17 Oct 19 2022 08:16:23

%S 1,0,3,8,33,120,451,1680,6273,23408,87363,326040,1216801,4541160,

%T 16947843,63250208,236052993,880961760,3287794051,12270214440,

%U 45793063713,170902040408,637815097923,2380358351280,8883618307201

%N a(n)=3*a(n-1)+3*a(n-2)-a(n-3);a(0)=1,a(1)=0,a(2)=3. a(n)=4*{a(n-1)+(-1)^n}-a(n-2);a(0)=1,a(1)=0.

%C For n>1, short leg of primitive Pythagorean triangles having an angle nearing pi/3 with larger values of sides.[Complete triple (X,Y,Z),X<Y<Z is given by X=a(n),Y=A001353(n),Z=A120893(n), with recurrence relations Y(i+1)=2*{Y(i)-(-1)^i} + 3*a(i) ; Z(i+1)=2*{2*Z(i)-a(i-1)} - 3*(-1)^i] A120893(n)=2*a(n)-(-1)^n.

%H Harvey P. Dale, <a href="/A120892/b120892.txt">Table of n, a(n) for n = 0..1000</a>

%H J. P. Chabert, <a href="http://jpm-chabert.club.fr/maths/Triangle_rect.htm">Right Triangle Applet (Hypotenuse & angles computation, given legs<350)</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, 3, -1).

%F Union of A045899 and A011922.

%F O.g.f.: -(-1+3*x)/((x+1)*(x^2-4*x+1)). - _R. J. Mathar_, Nov 23 2007

%t LinearRecurrence[{3,3,-1},{1,0,3},30] (* _Harvey P. Dale_, Mar 05 2014 *)

%o (PARI) a(n)=([0,1,0; 0,0,1; -1,3,3]^n*[1;0;3])[1,1] \\ _Charles R Greathouse IV_, Oct 19 2022

%K nonn,easy

%O 0,3

%A _Lekraj Beedassy_, Jul 13 2006

%E Corrected and extended by _T. D. Noe_, Nov 07 2006