A096383 Area of the Pythagorean triangle a = u^2 - v^2 (cf. A096382) when u=3, v=4,4,5,...

84, 240, 486, 840, 1320, 1944, 2730, 3696, 4860, 6240, 7854, 9720, 11856, 14280, 17010, 20064, 23460, 27216, 31350, 35880, 40824, 46200, 52026, 58320, 65100, 72384, 80190, 88536, 97440, 106920, 116994, 127680, 138996, 150960, 163590

Offset

2,1

Comments

When u = 1 except for the leading zeros, we get A007531. Since sides a,b of Pythagorean triple triangles are of opposite parity, the area will always be an integer.

Links

Table of n, a(n) for n=2..36.

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Formula

The area of a Pythagorean triangle of sides a < b < c is A = (1/2)*ab. Substituting a = u^2 - v^2, b = 2uv into A and simplifying, we get A = uv(v^2 - u^2).

a(n) = (n-3)*(n*3)*(n+3), n >= 3. - Zerinvary Lajos, Mar 05 2007

a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)- a(n-4); a(3)=84, a(4)=240, a(5)=486, a(6)=840. - Harvey P. Dale, Aug 28 2011

G.f.: (6*x*(5*x-16)+84)/(x-1)^4. - Harvey P. Dale, Aug 28 2011

Maple

seq((n-3)*(n*3)*(n+3), n=3..38); # Zerinvary Lajos, Mar 05 2007

Mathematica

Table[(n-3)(3n)(n+3), {n, 4, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {84, 240, 486, 840}, 40] (* Harvey P. Dale, Aug 28 2011 *)

Prog

(PARI) areagen(n, u) = for(v=u+1, n, print1(u*v*(v^2-u^2)", "))

Crossrefs

Cf. A007531.

Sequence in context: A219808 A219459 A112066 * A220043 A137210 A220009

Adjacent sequences: A096380 A096381 A096382 * A096384 A096385 A096386

Keyword

nonn,easy

Author

Cino Hilliard, Aug 05 2004

Status

approved