A334908 Area/6 of primitive Pythagorean triangles generated by {{2, 0}, {1, -1}}^n * {{2}, {1}}, for n >= 0.

1, 10, 220, 3080, 52976, 818720, 13333440, 211474560, 3398520576, 54257082880, 869067996160, 13897453373440, 222420341682176, 3558236809994240, 56935698394234880, 910939899548958720, 14575288593717067776, 233202615903456460800

Offset: 0

Author

Ralf Steiner, May 16 2020

Keywords

• nonn
• easy

Comments

Matrix {{2, 0}, {1, -1}} is [g_{-2}] given by Firstov in eq. (24).

These primitive Pythagorean triples are also given by Lee Price as (M_2)^n (3,4,5)^T (T for transposed), with M_2 = {{2, 1, 1}, {2, -2, 2}, {2, -1, 3}}.

For a primitive Pythagorean triangle (x, y, z) = (u^2-v^2, 2*u*v, u^2+v^2) the area is A = x*y/2 = u*v*(u^2 - v^2) = z*h/2 with altitude h, and h is an irreducible fraction. Here:

x(n) = A084175(n+2).

y(n) = 4*(A084175(n+1) - A084175(n)) = A054881(n+2).

= 2*A192382(n+1) = 4*A003683(n+1).

z(n) = A084175(n+2) + 2*A084175(n+1) - 4*A084175(n).

= A108924(n+2)/2 = A084175(n+2) + 2*A139818(n+1).

= A000302(n+1) + A139818(n+1).

u(n) = A000079(n+1) = 2^(n+1).

v(n) = A001045(n+1) = (2^(n+1) + (-1)^n)/3.

For the area A(n): Limit_{n -> oo} (3^3/(2^(4*n+7)))*A(n) = 1. See the formula section. - Wolfdieter Lang, Jun 14 2020

Examples

			a(0) = 3*4/12 = 1 for the triangle (3, 4, 5).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..825
• V. E. Firstov, A Special Matrix Transformation Semigroup of Primitive Pairs and the Genealogy of Pythagorean Triples; Mathematical Notes, volume 84, number 2, August 2008, pages 263-279; Link of the page (for the Russian article).
• H. Lee Price, The Pythagorean Tree: A New Species, arXiv:0809.4324 [math.HO], 2008-2011
• R. Steiner, Spezielle Folge primitiver pythagoräischer Dreiecke, researchgate.net, 2020
• Index entries for linear recurrences with constant coefficients, signature (10,120,-320,-1024).

Crossrefs

Cf. A000079, A000302, A001045, A003683, A054881, A084175, A108924, A139818, A192382.

Programs

Magma

[(2^(2*n+1)*(2^(2*n+5) -3) +(-2)^n*(3*2^(2*n+3) -1))/81: n in [0..40]]; // G. C. Greubel, Feb 18 2023

Mathematica

Table[(2^(2*n+1)*(2^(2*n+5) -3) + (-2)^n*(3*2^(2*n+3) -1))/3^4, {n,0,40}]

SageMath

[(2^(2*n+1)*(2^(2*n+5) -3) +(-2)^n*(3*2^(2*n+3) -1))/81 for n in range(41)] # G. C. Greubel, Feb 18 2023

Formula

a(n) = ( 2^(4*n+6) - 3*2^(2*n+1) - 3*(-2)^(3*n+3) - (-2)^n )/3^4.

G.f.: 1 / ((1 + 2*x)*(1 - 4*x)*(1 + 8*x)*(1 - 16*x)). - Colin Barker, Jun 11 2020

E.g.f.: (1/81)*(24*exp(-8*x) - exp(-2*x) - 6*exp(4*x) + 64*exp(16*x)). - G. C. Greubel, Feb 18 2023