A381005 Ordered short 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.

7, 175, 3007, 48895, 785407, 12578815, 201310207, 3221159935, 51539345407, 824632672255, 13194135339007, 211106215755775, 3377699653419007, 54043195260010495, 864691127381393407, 13835058050987196415, 221360928867334750207, 3541774862083514433535, 56668397794160864657407

Offset: 1

Author

Robert C. Lyons, Feb 12 2025

Keywords

• nonn
• easy

Comments

Proper subset of A020884.

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. A020884.

Cf. A381006 (long legs), A381007 (hypotenuses), A381008 (perimeters), A381009 (areas).

Programs

Magma

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

Mathematica

A381005[n_] := (3*# + 2)*(# - 2)/4 & [4^n]; Array[A381005, 20] (* or *)
LinearRecurrence[{21, -84, 64}, {7, 175, 3007}, 20] (* Paolo Xausa, Feb 26 2025 *)

PARI

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

Python

def A381005(n): return ((m:=1<<(n<<1)-1)-1)*(3*m+1) # Chai Wah Wu, Feb 13 2025

Formula

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

a(n) = sqrt( A381007(n)^2 - A381006(n)^2 ).

G.f.: (7 + 28*x - 80*x^2)/((1 - x)*(1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Feb 13 2025