A120892 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.

1, 0, 3, 8, 33, 120, 451, 1680, 6273, 23408, 87363, 326040, 1216801, 4541160, 16947843, 63250208, 236052993, 880961760, 3287794051, 12270214440, 45793063713, 170902040408, 637815097923, 2380358351280, 8883618307201

Offset: 0

Author

Lekraj Beedassy, Jul 13 2006

Keywords

• nonn
• easy

Comments

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),XA001353(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.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• J. P. Chabert, Right Triangle Applet (Hypotenuse & angles computation, given legs<350)
• Index entries for linear recurrences with constant coefficients, signature (3, 3, -1).

Programs

Mathematica

LinearRecurrence[{3,3,-1},{1,0,3},30] (* Harvey P. Dale, Mar 05 2014 *)

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

Formula

Union of A045899 and A011922.

O.g.f.: -(-1+3*x)/((x+1)*(x^2-4*x+1)). - R. J. Mathar, Nov 23 2007

Extensions

Corrected and extended by T. D. Noe, Nov 07 2006