This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A334908 #75 Mar 20 2025 22:31:18
%S A334908 1,10,220,3080,52976,818720,13333440,211474560,3398520576,54257082880,
%T A334908 869067996160,13897453373440,222420341682176,3558236809994240,
%U A334908 56935698394234880,910939899548958720,14575288593717067776,233202615903456460800
%N A334908 Area/6 of primitive Pythagorean triangles generated by {{2, 0}, {1, -1}}^n * {{2}, {1}}, for n >= 0.
%C A334908 Matrix {{2, 0}, {1, -1}} is [g_{-2}] given by Firstov in eq. (24).
%C A334908 These primitive Pythagorean triples are also given by Lee Price as (M_2)^n (3,4,5)^T (T for transposed), with M_2 = {{2, 1, 1}, {2, -2, 2}, {2, -1, 3}}.
%C A334908 For a primitive Pythagorean triangle (x, y, z) = (u^2-v^2, 2*u*v, u^2+v^2) the area is A = x*y/2 = u*v*(u^2 - v^2) = z*h/2 with altitude h, and h is an irreducible fraction. Here:
%C A334908 x(n) = A084175(n+2).
%C A334908 y(n) = 4*(A084175(n+1) - A084175(n)) = A054881(n+2).
%C A334908 = 2*A192382(n+1) = 4*A003683(n+1).
%C A334908 z(n) = A084175(n+2) + 2*A084175(n+1) - 4*A084175(n).
%C A334908 = A108924(n+2)/2 = A084175(n+2) + 2*A139818(n+1).
%C A334908 = A000302(n+1) + A139818(n+1).
%C A334908 u(n) = A000079(n+1) = 2^(n+1).
%C A334908 v(n) = A001045(n+1) = (2^(n+1) + (-1)^n)/3.
%C A334908 For the area A(n): Limit_{n -> oo} (3^3/(2^(4*n+7)))*A(n) = 1. See the formula section. - _Wolfdieter Lang_, Jun 14 2020
%H A334908 G. C. Greubel, <a href="/A334908/b334908.txt">Table of n, a(n) for n = 0..825</a>
%H A334908 V. E. Firstov, <a href="http://mi.mathnet.ru/eng/mz4074">A Special Matrix Transformation Semigroup of Primitive Pairs and the Genealogy of Pythagorean Triples</a>; Mathematical Notes, volume 84, number 2, August 2008, pages 263-279; Link of the page (for the Russian article).
%H A334908 H. Lee Price, <a href="http://arxiv.org/abs/0809.4324">The Pythagorean Tree: A New Species</a>, arXiv:0809.4324 [math.HO], 2008-2011
%H A334908 R. Steiner, <a href="https://www.researchgate.net/publication/340982803_Spezielle_Folge_primitiver_pythagoraischer_Dreiecke">Spezielle Folge primitiver pythagoräischer Dreiecke</a>, researchgate.net, 2020
%H A334908 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,120,-320,-1024).
%F A334908 a(n) = ( 2^(4*n+6) - 3*2^(2*n+1) - 3*(-2)^(3*n+3) - (-2)^n )/3^4.
%F A334908 G.f.: 1 / ((1 + 2*x)*(1 - 4*x)*(1 + 8*x)*(1 - 16*x)). - _Colin Barker_, Jun 11 2020
%F A334908 E.g.f.: (1/81)*(24*exp(-8*x) - exp(-2*x) - 6*exp(4*x) + 64*exp(16*x)). - _G. C. Greubel_, Feb 18 2023
%e A334908 a(0) = 3*4/12 = 1 for the triangle (3, 4, 5).
%t A334908 Table[(2^(2*n+1)*(2^(2*n+5) -3) + (-2)^n*(3*2^(2*n+3) -1))/3^4, {n,0,40}]
%o A334908 (Magma) [(2^(2*n+1)*(2^(2*n+5) -3) +(-2)^n*(3*2^(2*n+3) -1))/81: n in [0..40]]; // _G. C. Greubel_, Feb 18 2023
%o A334908 (SageMath) [(2^(2*n+1)*(2^(2*n+5) -3) +(-2)^n*(3*2^(2*n+3) -1))/81 for n in range(41)] # _G. C. Greubel_, Feb 18 2023
%Y A334908 Cf. A000079, A000302, A001045, A003683, A054881, A084175, A108924, A139818, A192382.
%K A334908 nonn,easy
%O A334908 0,2
%A A334908 _Ralf Steiner_, May 16 2020