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%I #52 Jan 29 2024 19:10:22
%S 0,4,20,120,696,4060,23660,137904,803760,4684660,27304196,159140520,
%T 927538920,5406093004,31509019100,183648021600,1070379110496,
%U 6238626641380,36361380737780,211929657785304,1235216565974040
%N Expansion of 4*x/((1+x)*(1-6*x+x^2)).
%C Related to Pythagorean triples: alternate terms of A001652 and A046090.
%C Even-valued legs of nearly isosceles right triangles: legs differ by 1. 0 is smaller leg of degenerate triangle with legs 0 and 1 and hypotenuse 1. - _Charlie Marion_, Nov 11 2003
%C The complete (nearly isosceles) primitive Pythagorean triple is given by {a(n), a(n)+(-1)^n, A001653(n)}. - _Lekraj Beedassy_, Feb 19 2004
%C Note also that A046092 is the even leg of this other class of nearly isosceles Pythagorean triangles {A005408(n), A046092(n), A001844(n)}, i.e., {2n+1, 2n(n+1), 2n(n+1)+1} where longer sides (viz. even leg and hypotenuse) are consecutive. - _Lekraj Beedassy_, Apr 22 2004
%C Union of even terms of A001652 and A046090. Sum of legs of primitive Pythagorean triangles is A002315(n) = 2*a(n) + (-1)^n. - _Lekraj Beedassy_, Apr 30 2004
%D A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.
%D W. Sierpiński, Pythagorean triangles, Dover Publications, Inc., Mineola, NY, 2003, p. 17. MR2002669.
%H G. C. Greubel, <a href="/A046729/b046729.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,-1).
%F a(n) = ((1+sqrt(2))^(2n+1) + (1-sqrt(2))^(2n+1) + 2*(-1)^(n+1))/4.
%F a(n) = A089499(n)*A089499(n+1).
%F a(n) = 4*A084158(n). - _Lekraj Beedassy_, Jul 16 2004
%F a(n) = ceiling((sqrt(2)+1)^(2*n+1) - (sqrt(2)-1)^(2*n+1) - 2*(-1)^n)/4. - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Nov 12 2004
%F a(n) is the k-th entry among the complete near-isosceles primitive Pythagorean triple A114336(n), where k = (3*(2n-1) - (-1)^n)/2, i.e., a(n) = A114336(A047235(n)), for positive n. - _Lekraj Beedassy_, Jun 04 2006
%F a(n) = A046727(n) - (-1)^n = 2*A114620(n). - _Lekraj Beedassy_, Aug 14 2006
%F From _George F. Johnson_, Aug 29 2012: (Start)
%F 2*a(n)*(a(n) + (-1)^n) + 1 = (A000129(2*n+1))^2;
%F n > 0, 2*a(n)*(a(n) + (-1)^n) + 1 = ((a(n+1) - a(n-1))/4)^2, a perfect square.
%F a(n+1) = (3*a(n) + 2*(-1)^n) + 2*sqrt(2*a(n)*(a(n) + (-1)^n)+ 1).
%F a(n-1) = (3*a(n) + 2*(-1)^n) - 2*sqrt(2*a(n)*(a(n) + (-1)^n)+ 1).
%F a(n+1) = 6*a(n) - a(n-1) + 4*(-1)^n.
%F a(n+1) = 5*a(n) + 5*a(n-1) - a(n-2).
%F a(n+1) *a(n-1) = a(n)*(a(n) + 4*(-1)^n).
%F a(n) = (sqrt(1 + 8*A029549(n)) - (-1)^n)/2.
%F a(n) = A002315(n) - A084159(n) = A084159(n) - (-1)^n.
%F a(n) = A001652(n) + (1 - (-1)^n)/2 = A046090(n) - (1 + (-1)^n)/2.
%F Limit_{n->oo} a(n)/a(n-1) = 3 + 2*sqrt(2).
%F Limit_{n->oo} a(n)/a(n-2) = 17 + 12*sqrt(2).
%F Limit_{n->oo} a(n)/a(n-r) = (3 + 2*sqrt(2))^r.
%F Limit_{n->oo} a(n-r)/a(n) = (3 - 2*sqrt(2))^r. (End)
%F From _G. C. Greubel_, Feb 11 2023: (Start)
%F a(n) = (A001333(2*n+1) - 2*(-1)^n)/4.
%F a(n) = (1/2)*(A001109(n+1) + A001109(n) - (-1)^n). (End)
%e [1,0,1]*[1,2,2; 2,1,2; 2,2,3]^0 gives (degenerate) primitive Pythagorean triple [1, 0, 1], so a(0) = 0. [1,0,1]*[1,2,2; 2,1,2; 2,2,3]^7 gives primitive Pythagorean triple [137903, 137904, 195025] so a(7) = 137904.
%e G.f. = 4*x + 20*x^2 + 120*x^3 + 696*x^4 + 4060*x^5 + 23660*x^6 + ...
%t LinearRecurrence[{5,5,-1}, {0,4,20}, 25] (* _Vincenzo Librandi_, Jul 29 2019 *)
%o (PARI) a(n)=n%2+(real((1+quadgen(8))^(2*n+1))-1)/2
%o (PARI) a(n)=if(n<0,-a(-1-n),polcoeff(4*x/(1+x)/(1-6*x+x^2)+x*O(x^n),n))
%o (Magma) [4*Floor(((Sqrt(2)+1)^(2*n+1)-(Sqrt(2)-1)^(2*n+1)-2*(-1)^n) / 16): n in [0..30]]; // _Vincenzo Librandi_, Jul 29 2019
%o (SageMath) [(lucas_number2(2*n+1,2,-1) -2*(-1)^n)/4 for n in range(41)] # _G. C. Greubel_, Feb 11 2023
%Y Cf. A000129, A001109, A001333, A001652, A002315, A029549, A046090.
%Y Cf. A046727, A047235, A084158, A084159, A089499, A114336.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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