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%I #23 Feb 20 2024 09:23:22
%S 1,2,5,7,19,13,71,97,265,181,989,1351,3691,2521,13775,18817,51409,
%T 35113,191861,262087,716035,489061,2672279,3650401,9973081,6811741,
%U 37220045,50843527,138907099,94875313,518408351,708158977,1934726305,1321442641
%N Numerators of fractions x^n + y^n, where x + y = 1 and x^2 + y^2 = 2.
%C x and y are given by -A152422 and 1-A152422. - _R. J. Mathar_, Mar 01 2010
%C 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
%H Vincenzo Librandi, <a href="/A173299/b173299.txt">Table of n, a(n) for n = 1..200</a>
%F a(n) = numerator of ((1 + sqrt(3))/2)^n + ((1 - sqrt(3))/2)^n.
%e a(3) = 5 because x^3 + y^3 is 2.5 and 2.5 is 5/2.
%p 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
%t 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 *)
%o (PARI) { a(n) = numerator( 2 * polcoeff( lift( Mod((1+x)/2,x^2-3)^n ), 0) ) }
%o (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
%o (Python)
%o from fractions import Fraction
%o def a173299_gen(a, b):
%o while True:
%o yield a.numerator
%o b, a = b + Fraction(a, 2), b
%o g = a173299_gen(1, 2)
%o print([next(g) for _ in range(34)]) # _Nick Hobson_, Feb 20 2024
%Y Cf. A173300 (denominators).
%K nonn,frac
%O 1,2
%A _J. Lowell_, Feb 15 2010
%E Formula, more terms, and PARI script from _Max Alekseyev_, Feb 24 2010
%E More terms from _Klaus Brockhaus_ and _R. J. Mathar_, Mar 01 2010
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