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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.

%I A381006 #39 May 06 2025 11:31:46
%S A381006 24,288,4224,66048,1050624,16785408,268468224,4295098368,68720001024,
%T A381006 1099513724928,17592194433024,281475010265088,4503599761588224,
%U A381006 72057594574798848,1152921506754330624,18446744082299486208,295147905213712564224,4722366483007084167168
%N 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.
%C A381006 Proper subset of A020883.
%C A381006 Conjecture: These Pythagorean triangles are primitive. Verified up to n=100000.
%C A381006 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
%H A381006 Paolo Xausa, <a href="/A381006/b381006.txt">Table of n, a(n) for n = 1..800</a>
%H A381006 John D. Cook, <a href="https://www.johndcook.com/blog/2025/02/11/binary-triples/">Sparse binary Pythagorean triples</a> (2025).
%H A381006 H. S. Uhler, <a href="https://www.jstor.org/stable/2306207">A Colossal Primitive Pythagorean Triangle</a>, The American Mathematical Monthly, Vol. 57, No. 5 (May, 1950), pp. 331-332.
%H A381006 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pythagorean_triple">Pythagorean triple</a>.
%H A381006 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-64).
%F A381006 a(n) = 2^(4n) + 2^(2n+1).
%F A381006 a(n) = sqrt( A381007(n)^2 - A381005(n)^2 ).
%F A381006 G.f.: 24*(1 - 8*x)/((1 - 4*x)*(1 - 16*x)). - _Stefano Spezia_, Feb 13 2025
%t A381006 A381006[n_] := #*(# + 2) & [4^n]; Array[A381006, 20] (* or *)
%t A381006 LinearRecurrence[{20, -64}, {24, 288}, 20] (* _Paolo Xausa_, Feb 26 2025 *)
%o A381006 (PARI) a(n) = 2^(4*n) + 2^(2*n+1)
%o A381006 (Magma) [2^(4*n) + 2^(2*n+1): n in [1..20]];
%o A381006 (Python)
%o A381006 def A381006(n): return (m:=1<<(n<<1))*(m+2) # _Chai Wah Wu_, Feb 13 2025
%Y A381006 Cf. A020883.
%Y A381006 Cf. A381005 (short legs), A381007 (hypotenuses), A381008 (perimeters), A381009 (areas).
%K A381006 nonn,easy
%O A381006 1,1
%A A381006 _Robert C. Lyons_, Feb 12 2025