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

1,2

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

Links

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

Formula

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

Example

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

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 *)

Prog

(PARI) { a(n) = numerator( 2 * polcoeff( lift( Mod((1+x)/2, x^2-3)^n ), 0) ) }
(Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=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
(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

Crossrefs

Cf. A173300 (denominators).

Sequence in context: A027038 A341322 A173929 * A097052 A102937 A045358

Adjacent sequences: A173296 A173297 A173298 * A173300 A173301 A173302

Keyword

nonn,frac

Author

J. Lowell, Feb 15 2010

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

Status

approved