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, 33154114877520, 123732841202883, 461777249934008

Offset

0,3

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),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]

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). [Dead Link]

Luke Underwood, Right Triangle Maker, GeoGebra.

Index entries for linear recurrences with constant coefficients, signature (3,3,-1).

Formula

2*a(n)-(-1)^n = A120893(n).

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

From Amiram Eldar, Jan 26 2026: (Start)

Sum_{n>=2} 1/a(n) = 1/2.

Sum_{n>=2} (-1)^n/a(n) = sqrt(3) - 3/2. (End)

Mathematica

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

Prog

(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

Crossrefs

Union of A045899 and A011922.

Cf. A001353, A120893.

Sequence in context: A148916 A148917 A195499 * A225688 A109655 A184255

Adjacent sequences: A120889 A120890 A120891 * A120893 A120894 A120895

Keyword

nonn,easy

Author

Lekraj Beedassy, Jul 13 2006

Extensions

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

Status

approved