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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.

%I A381005 #37 May 06 2025 11:31:36
%S A381005 7,175,3007,48895,785407,12578815,201310207,3221159935,51539345407,
%T A381005 824632672255,13194135339007,211106215755775,3377699653419007,
%U A381005 54043195260010495,864691127381393407,13835058050987196415,221360928867334750207,3541774862083514433535,56668397794160864657407
%N 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.
%C A381005 Proper subset of A020884.
%C A381005 Conjecture: These Pythagorean triangles are primitive. Verified up to n=100000.
%C A381005 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 A381005 Paolo Xausa, <a href="/A381005/b381005.txt">Table of n, a(n) for n = 1..800</a>
%H A381005 John D. Cook, <a href="https://www.johndcook.com/blog/2025/02/11/binary-triples/">Sparse binary Pythagorean triples</a> (2025).
%H A381005 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 A381005 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pythagorean_triple">Pythagorean triple</a>.
%H A381005 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (21,-84,64).
%F A381005 a(n) = 2^(4n) - 2^(4n-2) - 2^(2n) - 1.
%F A381005 a(n) = sqrt( A381007(n)^2 - A381006(n)^2 ).
%F A381005 G.f.: (7 + 28*x - 80*x^2)/((1 - x)*(1 - 4*x)*(1 - 16*x)). - _Stefano Spezia_, Feb 13 2025
%t A381005 A381005[n_] := (3*# + 2)*(# - 2)/4 & [4^n]; Array[A381005, 20] (* or *)
%t A381005 LinearRecurrence[{21, -84, 64}, {7, 175, 3007}, 20] (* _Paolo Xausa_, Feb 26 2025 *)
%o A381005 (PARI) a(n) = 2^(4*n) - 2^(4*n-2) - 2^(2*n) - 1
%o A381005 (Magma) [2^(4*n) - 2^(4*n-2) - 2^(2*n) - 1: n in [1..20]];
%o A381005 (Python)
%o A381005 def A381005(n): return ((m:=1<<(n<<1)-1)-1)*(3*m+1) # _Chai Wah Wu_, Feb 13 2025
%Y A381005 Cf. A020884.
%Y A381005 Cf. A381006 (long legs), A381007 (hypotenuses), A381008 (perimeters), A381009 (areas).
%K A381005 nonn,easy
%O A381005 1,1
%A A381005 _Robert C. Lyons_, Feb 12 2025