A381006 Ordered long legs of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

24, 288, 4224, 66048, 1050624, 16785408, 268468224, 4295098368, 68720001024, 1099513724928, 17592194433024, 281475010265088, 4503599761588224, 72057594574798848, 1152921506754330624, 18446744082299486208, 295147905213712564224, 4722366483007084167168

Offset: 1

Author

Robert C. Lyons, Feb 12 2025

Keywords

• nonn
• easy

Comments

Proper subset of A020883.

Conjecture: These Pythagorean triangles are primitive. Verified up to n=100000.

The preceding conjecture is true, since, for n>=1, the values of a,b,c are given by Euclid's formula for generating Pythagorean triples: a=2xy, b=x^2-y^2, c=x^2+y^2 with x=2^(2n) and y=2^(2n-1)+1 and x and y are coprime and x is even and y is odd. - Chai Wah Wu, Feb 13 2025

Links

• Paolo Xausa, Table of n, a(n) for n = 1..800
• John D. Cook, Sparse binary Pythagorean triples (2025).
• H. S. Uhler, A Colossal Primitive Pythagorean Triangle, The American Mathematical Monthly, Vol. 57, No. 5 (May, 1950), pp. 331-332.
• Wikipedia, Pythagorean triple.
• Index entries for linear recurrences with constant coefficients, signature (20,-64).

Crossrefs

Cf. A020883.

Cf. A381005 (short legs), A381007 (hypotenuses), A381008 (perimeters), A381009 (areas).

Programs

Magma

[2^(4*n) + 2^(2*n+1): n in [1..20]];

Mathematica

A381006[n_] := #*(# + 2) & [4^n]; Array[A381006, 20] (* or *)
LinearRecurrence[{20, -64}, {24, 288}, 20] (* Paolo Xausa, Feb 26 2025 *)

PARI

a(n) = 2^(4*n) + 2^(2*n+1)

Python

def A381006(n): return (m:=1<<(n<<1))*(m+2) # Chai Wah Wu, Feb 13 2025

Formula

a(n) = 2^(4n) + 2^(2n+1).

a(n) = sqrt( A381007(n)^2 - A381005(n)^2 ).

G.f.: 24*(1 - 8*x)/((1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Feb 13 2025