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A381008 Ordered perimeters 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 A381008 #21 May 06 2025 11:32:08
%S A381008 56,800,12416,197120,3147776,50339840,805339136,12885032960,
%T A381008 206158954496,3298536980480,52776566521856,844424963686400,
%U A381008 13510799016329216,216172782650654720,3458764515968024576,55340232229718589440,885443715572418215936,14167099448746374594560
%N A381008 Ordered perimeters 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 A381008 Proper subset of A024364.
%H A381008 Paolo Xausa, <a href="/A381008/b381008.txt">Table of n, a(n) for n = 1..800</a>
%H A381008 John D. Cook, <a href="https://www.johndcook.com/blog/2025/02/11/binary-triples/">Sparse binary Pythagorean triples</a> (2025).
%H A381008 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 A381008 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pythagorean_triple">Pythagorean triple</a>.
%H A381008 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-64).
%F A381008 a(n) = A381005(n) + A381006(n) + A381007(n).
%F A381008 a(n) = 2^(4n+1) + 2^(2n+1) + 2^(4n).
%F A381008 G.f.: 8*(7 - 40*x)/((1 - 4*x)*(1 - 16*x)). - _Stefano Spezia_, Feb 13 2025
%t A381008 A381008[n_] := #*(3*# + 2) & [4^n]; Array[A381008, 20] (* or *)
%t A381008 LinearRecurrence[{20, -64}, {56, 800}, 20] (* _Paolo Xausa_, Feb 26 2025 *)
%o A381008 (PARI) a(n) = 2^(4*n+1) + 2^(2*n+1) + 2^(4*n)
%o A381008 (Magma) [2^(4*n+1) + 2^(2*n+1) + 2^(4*n): n in [1..20]];
%o A381008 (Python)
%o A381008 def A381008(n): return (m:=1<<(n<<1))*(2+3*m) # _Chai Wah Wu_, Feb 13 2025
%Y A381008 Cf. A024364.
%Y A381008 Cf. A381005 (short legs), A381006 (long legs), A381007 (hypotenuses), A381009 (areas).
%K A381008 nonn,easy
%O A381008 1,1
%A A381008 _Robert C. Lyons_, Feb 12 2025