A195549 Hypotenuses of primitive Pythagorean triples in A195547 and A195548.

1, 5, 13, 17, 89, 233, 305, 1597, 4181, 5473, 28657, 75025, 98209, 514229, 1346269, 1762289, 9227465, 24157817, 31622993, 165580141, 433494437, 567451585, 2971215073, 7778742049, 10182505537, 53316291173, 139583862445, 182717648081

Offset: 1

Author

Clark Kimberling, Sep 20 2011

Keywords

• nonn

Comments

See A195500 for discussion and list of related sequences; see A195547 for Mathematica program.

Crossrefs

Cf. A000045, A195500, A195547, A195548, A131534.

Programs

Maple

with(combinat):f:= n-> fibonacci(n):seq((f(n)^2+f(n+1)^2)/(2-((n+2)^2 mod 3)), n=1..25);

Formula

a(n) = (F(n)^2 + F(n+1)^2)/(2 - ((n+2)^2 mod 3)), F(n) = Fibonacci(n). - Gary Detlefs, Oct 14 2011

Conjectures from Colin Barker, Apr 08 2012: (Start)

a(n) = 18*a(n-3) - a(n-6).

G.f.: x*(1+5*x+13*x^2-x^3-x^4-x^5)/((1-3*x+x^2)*(1+3*x+8*x^2+3*x^3+x^4)). (End)

Conjecture: a(n) is the denominator of the reduced fraction (F(2*n+1)-2)/F(2*n+1), F(n) = Fibonacci(n). - Sébastien Labbé, May 06 2022