A334638 -id:A334638 - cp's OEIS frontend

A334909 Area/6 of primitive Pythagorean triangles given in A334638 as triples.

1, 35, 770, 14260, 244776, 4053840, 65979040, 1064678720, 17107266176, 274296689920, 4393395202560, 70331527418880, 1125602147608576, 18012016334950400, 288211318352814080, 4611533554425610240, 73785756576381566976, 1180581862943988449280

Offset: 0

Ralf Steiner, May 16 2020

See A334638 for these triangles, and also for the Firstov reference.
For 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.
From A334638 follows A(n)/6 = (x(n)/3)*(y(n)/4) = A171477(n)*A010036(n), for n >= 0. See the formula section.
Limit_{n->infinity} A(n)/(3*2^(4*n+3)) = 1. See the formula section. - Wolfdieter Lang, Jun 14 2020

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

Index entries for linear recurrences with constant coefficients, signature (30,-280,960,-1024).

Cf. A010036, A171477, A334638.

Table[ 2^(-1 + n) (-1 + 3 2^n) (-1 + 2^(1 + n)) (-1 + 2^(2 + n))/3, {n, 0, 17}]

Vec((1 + 5*x) / ((1 - 2*x)*(1 - 4*x)*(1 - 8*x)*(1 - 16*x)) + O(x^20)) \\ Colin Barker, May 17 2020

a(n) = 2^(n-1)*(3*2^n - 1)*(2^(n+1) - 1)*(2^(n+2) - 1)/3.
a(n) = 2^(4*n+2)*(1 - 13/(3*2^(n+2)) + 3/2^(2*n+3) - 1/(3*2^(3*(n+1)))), for n >= 0.
From Colin Barker: (Start)
G.f.: (1 + 5*x) / ((1 - 2*x)*(1 - 4*x)*(1 - 8*x)*(1 - 16*x)).
a(n) = 30*a(n-1) - 280*a(n-2) + 960*a(n-3) - 1024*a(n-4) for n > 3. (End)

Edited by Wolfdieter Lang, Jun 14 2020

cp's OEIS Frontend

A334909 Area/6 of primitive Pythagorean triangles given in A334638 as triples.

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