cp's OEIS Frontend

A381009 Ordered areas 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 A381009 #19 May 06 2025 11:32:14
%S A381009 84,25200,6350784,1614708480,412583721984,105570270965760,
%T A381009 27022696873181184,6917599389942743040,1770891934572664848384,
%U A381009 453347470584212823736320,116056897129722086198083584,29710562123440325102508441600,7605903676927233379495034486784,1947111326786263531071061496954880
%N A381009 Ordered areas 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 A381009 Proper subset of A024406.
%H A381009 Paolo Xausa, <a href="/A381009/b381009.txt">Table of n, a(n) for n = 1..400</a>
%H A381009 John D. Cook, <a href="https://www.johndcook.com/blog/2025/02/11/binary-triples/">Sparse binary Pythagorean triples</a> (2025).
%H A381009 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 A381009 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pythagorean_triple">Pythagorean triple</a>.
%H A381009 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (340,-22848,348160,-1048576).
%F A381009 a(n) = A381005(n) * A381006(n) / 2.
%F A381009 a(n) = (2^(4n) + 2^(2n+1)) * (2^(4n) - 2^(4n-2) - 2^(2n) - 1) / 2.
%F A381009 G.f.: 12*(7 - 280*x - 24832*x^2 + 163840*x^3)/((1 - 4*x)*(1 - 16*x)*(1 - 64*x)*(1 - 256*x)). - _Stefano Spezia_, Feb 13 2025
%t A381009 A381009[n_] := (3*# + 2)*(# + 2)*(# - 2)*2^(2*n - 3) & [4^n]; Array[A381009, 20] (* or *)
%t A381009 LinearRecurrence[{340, -22848, 348160, -1048576}, {84, 25200, 6350784, 1614708480}, 20] (* _Paolo Xausa_, Feb 26 2025 *)
%o A381009 (PARI) a(n) = (2^(4*n) + 2^(2*n+1)) * (2^(4*n) - 2^(4*n-2) - 2^(2*n) - 1) / 2
%o A381009 (Magma) [(2^(4*n) + 2^(2*n+1)) * (2^(4*n) - 2^(4*n-2) - 2^(2*n) - 1) / 2: n in [1..20]];
%o A381009 (Python)
%o A381009 def A381009(n): return (m:=1<<(n<<1)-1)*(m-1)*(m+1)*(3*m+1)<<1 # _Chai Wah Wu_, Feb 13 2025
%Y A381009 Cf. A024406.
%Y A381009 Cf. A381005 (short legs), A381006 (long legs), A381007 (hypotenuses), A381008 (perimeters).
%K A381009 nonn,easy
%O A381009 1,1
%A A381009 _Robert C. Lyons_, Feb 12 2025