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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.

%I A381007 #34 May 06 2025 15:37:39
%S A381007 25,337,5185,82177,1311745,20975617,335560705,5368774657,85899608065,
%T A381007 1374390583297,21990236749825,351843737665537,5629499601321985,
%U A381007 90071992815845377,1441151881832300545,23058430096431906817,368934881491370901505,5902958103655775993857
%N 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.
%C A381007 Proper subset of A020882.
%C A381007 Conjecture: These Pythagorean triangles are primitive. Verified up to n=100000.
%C A381007 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 A381007 Paolo Xausa, <a href="/A381007/b381007.txt">Table of n, a(n) for n = 1..800</a>
%H A381007 John D. Cook, <a href="https://www.johndcook.com/blog/2025/02/11/binary-triples/">Sparse binary Pythagorean triples</a> (2025).
%H A381007 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 A381007 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pythagorean_triple">Pythagorean triple</a>.
%H A381007 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (21,-84,64).
%F A381007 a(n) = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.
%F A381007 a(n) = sqrt( A381005(n)^2 + A381006(n)^2 ).
%F A381007 G.f.: (25 - 188*x + 208*x^2)/((1 - x)*(1 - 4*x)*(1 - 16*x)). - _Stefano Spezia_, Feb 13 2025
%t A381007 A381007[n_] := 5*4^(2*n - 1) + 4^n + 1; Array[A381007, 20] (* or *)
%t A381007 LinearRecurrence[{21, -84, 64}, {25, 337, 5185}, 20] (* _Paolo Xausa_, Feb 26 2025 *)
%o A381007 (PARI) a(n) = 2^(4*n) + 2^(4*n-2) + 2^(2*n) + 1
%o A381007 (Magma) [2^(4*n) + 2^(4*n-2) + 2^(2*n) + 1: n in [1..20]];
%o A381007 (Python)
%o A381007 def A381007(n): return (m:=1<<(n<<1)-1)*(5*m+2)+1 # _Chai Wah Wu_, Feb 13 2025
%Y A381007 Cf. A020882.
%Y A381007 Cf. A381005 (short legs), A381006 (long legs), A381008 (perimeters), A381009 (areas).
%K A381007 nonn,easy
%O A381007 1,1
%A A381007 _Robert C. Lyons_, Feb 12 2025