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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

(list; graph; refs; listen; history; text; internal format)

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