A381007 Ordered hypotenuses 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.

25, 337, 5185, 82177, 1311745, 20975617, 335560705, 5368774657, 85899608065, 1374390583297, 21990236749825, 351843737665537, 5629499601321985, 90071992815845377, 1441151881832300545, 23058430096431906817, 368934881491370901505, 5902958103655775993857

Offset: 1

Author

Robert C. Lyons, Feb 12 2025

Keywords

• nonn
• easy

Comments

Proper subset of A020882.

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 (21,-84,64).

Crossrefs

Cf. A020882.

Cf. A381005 (short legs), A381006 (long legs), A381008 (perimeters), A381009 (areas).

Programs

Magma

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

Mathematica

A381007[n_] := 5*4^(2*n - 1) + 4^n + 1; Array[A381007, 20] (* or *)
LinearRecurrence[{21, -84, 64}, {25, 337, 5185}, 20] (* Paolo Xausa, Feb 26 2025 *)

PARI

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

Python

def A381007(n): return (m:=1<<(n<<1)-1)*(5*m+2)+1 # Chai Wah Wu, Feb 13 2025

Formula

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

a(n) = sqrt( A381005(n)^2 + A381006(n)^2 ).

G.f.: (25 - 188*x + 208*x^2)/((1 - x)*(1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Feb 13 2025