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 A334909 #59 Dec 21 2024 01:03:08
%S A334909 1,35,770,14260,244776,4053840,65979040,1064678720,17107266176,
%T A334909 274296689920,4393395202560,70331527418880,1125602147608576,
%U A334909 18012016334950400,288211318352814080,4611533554425610240,73785756576381566976,1180581862943988449280
%N A334909 Area/6 of primitive Pythagorean triangles given in A334638 as triples.
%C A334909 See A334638 for these triangles, and also for the Firstov reference.
%C A334909 For 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.
%C A334909 From A334638 follows A(n)/6 = (x(n)/3)*(y(n)/4) = A171477(n)*A010036(n), for n >= 0. See the formula section.
%C A334909 Limit_{n->infinity} A(n)/(3*2^(4*n+3)) = 1. See the formula section. - _Wolfdieter Lang_, Jun 14 2020
%H A334909 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (30,-280,960,-1024).
%F A334909 a(n) = 2^(n-1)*(3*2^n - 1)*(2^(n+1) - 1)*(2^(n+2) - 1)/3.
%F A334909 a(n) = 2^(4*n+2)*(1 - 13/(3*2^(n+2)) + 3/2^(2*n+3) - 1/(3*2^(3*(n+1)))), for n >= 0.
%F A334909 From _Colin Barker_: (Start)
%F A334909 G.f.: (1 + 5*x) / ((1 - 2*x)*(1 - 4*x)*(1 - 8*x)*(1 - 16*x)).
%F A334909 a(n) = 30*a(n-1) - 280*a(n-2) + 960*a(n-3) - 1024*a(n-4) for n > 3. (End)
%e A334909 a(0) = 3*4/12 = 1 for (3, 4, 5).
%t A334909 Table[ 2^(-1 + n) (-1 + 3 2^n) (-1 + 2^(1 + n)) (-1 + 2^(2 + n))/3, {n, 0, 17}]
%o A334909 (PARI) Vec((1 + 5*x) / ((1 - 2*x)*(1 - 4*x)*(1 - 8*x)*(1 - 16*x)) + O(x^20)) \\ Colin Barker, May 17 2020
%Y A334909 Cf. A010036, A171477, A334638.
%K A334909 nonn,easy
%O A334909 0,2
%A A334909 _Ralf Steiner_, May 16 2020
%E A334909 Edited by _Wolfdieter Lang_, Jun 14 2020