A381008 Ordered perimeters 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.

56, 800, 12416, 197120, 3147776, 50339840, 805339136, 12885032960, 206158954496, 3298536980480, 52776566521856, 844424963686400, 13510799016329216, 216172782650654720, 3458764515968024576, 55340232229718589440, 885443715572418215936, 14167099448746374594560

Offset: 1

Author

Robert C. Lyons, Feb 12 2025

Keywords

• nonn
• easy

Comments

Proper subset of A024364.

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. A024364.

Cf. A381005 (short legs), A381006 (long legs), A381007 (hypotenuses), A381009 (areas).

Programs

Magma

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

Mathematica

A381008[n_] := #*(3*# + 2) & [4^n]; Array[A381008, 20] (* or *)
LinearRecurrence[{20, -64}, {56, 800}, 20] (* Paolo Xausa, Feb 26 2025 *)

PARI

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

Python

def A381008(n): return (m:=1<<(n<<1))*(2+3*m) # Chai Wah Wu, Feb 13 2025

Formula

a(n) = A381005(n) + A381006(n) + A381007(n).

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

G.f.: 8*(7 - 40*x)/((1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Feb 13 2025