%I A046729
%S A046729 0,4,20,120,696,4060,23660,137904,803760,4684660,27304196,159140520,
%T A046729 927538920,5406093004,31509019100,183648021600,1070379110496,
%U A046729 6238626641380,36361380737780,211929657785304,1235216565974040
%N A046729 G.f.: 4x/((1+x)(1-6x+x^2)).
%C A046729 Related to Pythagorean triples: alternate terms of A001652 and A046090.
%C A046729 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 (charliem(AT)bestweb.net), Nov 11 
               2003
%C A046729 The complete (nearly isosceles) primitive Pythagorean triple is given 
               by {a(n),a(n)+(-1)^n, A001653(n)}. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Feb 19 2004
%C A046729 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 hypotenus) are consecutive. - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Apr 22 2004
%C A046729 Union of even entries of A001652 and A046090. Sum of legs of primitive 
               Pythagorean triangles is A002315(n)=2*a(n)+(-1)^n. - Lekraj Beedassy 
               (blekraj(AT)yahoo.com), Apr 30 2004
%D A046729 A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, 
               pp. 122-125, 1964.
%D A046729 W. Sierpinski, Pythagorean triangles, Dover Publications, Inc., Mineola, 
               NY, 2003, p. 17. MR2002669.
%H A046729 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A046729 a(n)=((1+sqrt(2))^(2n+1)+(1-sqrt(2))^(2n+1)+2(-1)^(n+1))/4; a(n)=A089499(n)*A089499(n+1); 
               cf. A084159.
%F A046729 a(n)=4*A084158(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 16 2004
%F A046729 a(n) = ceil((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 A046729 a(n) is the k-th entry amongst 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 (blekraj(AT)yahoo.com), 
               Jun 04 2006
%F A046729 a(n) = A046727(n)-(-1)^n = 2*A114620(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), 
               Aug 14 2006
%e A046729 [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
%o A046729 (PARI) a(n)=n%2+(real((1+quadgen(8))^(2*n+1))-1)/2
%o A046729 (PARI) a(n)=if(n<0,-a(-1-n),polcoeff(4*x/(1+x)/(1-6*x+x^2)+x*O(x^n),n))
%Y A046729 Cf. A046727, A084159, A084158, A001652, A046090.
%Y A046729 Sequence in context: A101055 A013197 A089498 this_sequence A093123 A092055 
               A001715
%Y A046729 Adjacent sequences: A046726 A046727 A046728 this_sequence A046730 A046731 
               A046732
%K A046729 nonn,easy
%O A046729 0,2
%A A046729 N. J. A. Sloane (njas(AT)research.att.com).
%E A046729 More terms from Philip Sung (phil(AT)main.nu), May 05 2001