A173299 Numerators of fractions x^n + y^n, where x + y = 1 and x^2 + y^2 = 2.

1, 2, 5, 7, 19, 13, 71, 97, 265, 181, 989, 1351, 3691, 2521, 13775, 18817, 51409, 35113, 191861, 262087, 716035, 489061, 2672279, 3650401, 9973081, 6811741, 37220045, 50843527, 138907099, 94875313, 518408351, 708158977, 1934726305, 1321442641

Offset: 1

Author

J. Lowell, Feb 15 2010

Keywords

• nonn
• frac

Comments

x and y are given by -A152422 and 1-A152422. - R. J. Mathar, Mar 01 2010

Letting f(n) = x^n + y^n, recurrence relation f(n) = f(n - 1) + f(n - 2)/2 implies a(n) / A173300(n) = A026150(n) / 2^(n - 1). - Nick Hobson, Jan 30 2024

Examples

			a(3) = 5 because x^3 + y^3 is 2.5 and 2.5 is 5/2.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A173300 (denominators).

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(2*x^2-2*x-1); S:=[ r^n+(1-r)^n: n in [1..34] ]; [ Numerator(RationalField()!S[j]): j in [1..#S] ]; // Klaus Brockhaus, Mar 02 2010

Maple

A173299 := proc(n) local x,y ; x := (1+sqrt(3))/2 ; y := (1-sqrt(3))/2 ; expand(x^n+y^n) ; numer(%) ; end proc: # R. J. Mathar, Mar 01 2010

Mathematica

Module[{x=(1-Sqrt[3])/2,y},y=1-x;Table[x^n+y^n,{n,40}]]//Simplify// Numerator (* Harvey P. Dale, Aug 24 2019 *)

PARI

{ a(n) = numerator( 2 * polcoeff( lift( Mod((1+x)/2,x^2-3)^n ), 0) ) }

Python

from fractions import Fraction
def a173299_gen(a, b):
    while True:
        yield a.numerator
        b, a = b + Fraction(a, 2), b
g = a173299_gen(1, 2)
print([next(g) for  in range(34)])  # _Nick Hobson, Feb 20 2024

Formula

a(n) = numerator of ((1 + sqrt(3))/2)^n + ((1 - sqrt(3))/2)^n.

Extensions

Formula, more terms, and PARI script from Max Alekseyev, Feb 24 2010

More terms from Klaus Brockhaus and R. J. Mathar, Mar 01 2010