0,2

Related to Pythagorean triples: alternate terms of A001652 and A046090.

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

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

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

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

A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 122-125, 1964.

W. Sierpinski, Pythagorean triangles, Dover Publications, Inc., Mineola, NY, 2003, p. 17. MR2002669.

Index entries for sequences related to linear recurrences with constant coefficients

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.

a(n)=4*A084158(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 16 2004

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

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

a(n) = A046727(n)-(-1)^n = 2*A114620(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Aug 14 2006

[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

(PARI) a(n)=n%2+(real((1+quadgen(8))^(2*n+1))-1)/2

(PARI) a(n)=if(n<0, -a(-1-n), polcoeff(4*x/(1+x)/(1-6*x+x^2)+x*O(x^n), n))

Cf. A046727, A084159, A084158, A001652, A046090.

Sequence in context: A101055 A013197 A089498 this_sequence A093123 A092055 A001715

Adjacent sequences: A046726 A046727 A046728 this_sequence A046730 A046731 A046732

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Philip Sung (phil(AT)main.nu), May 05 2001