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A173299 Numerators of fractions x^n + y^n, where x + y = 1 and x^2 + y^2 = 2. 4
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 (list; graph; refs; listen; history; text; internal format)
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
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
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

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Last modified May 21 08:56 EDT 2024. Contains 372733 sequences. (Running on oeis4.)