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A173300 a(n) is the denominator of the fraction f = x^n + y^n given that x + y = 1 and x^2 + y^2 = 2.

%I A173300 #29 Feb 04 2024 18:39:09
%S A173300 1,1,2,2,4,2,8,8,16,8,32,32,64,32,128,128,256,128,512,512,1024,512,
%T A173300 2048,2048,4096,2048,8192,8192,16384,8192,32768,32768,65536,32768,
%U A173300 131072,131072,262144,131072,524288,524288,1048576,524288,2097152,2097152,4194304,2097152
%N A173300 a(n) is the denominator of the fraction f = x^n + y^n given that x + y = 1 and x^2 + y^2 = 2.
%C A173300 The denominators of the coefficients of the Taylor series representation of (1+x)/(1-2*x-11*x^2-6*x^3) around x=-1 lead to this sequence, see the Maple program. - _Johannes W. Meijer_, Aug 16 2010
%F A173300 a(n) = denominator of ((1+sqrt(3))/2)^n + ((1-sqrt(3))/2)^n. - _Max Alekseyev_, Feb 23 2010
%F A173300 Conjecture: a(n) = 4*a(n-4), for n >= 7. - _Paolo Xausa_, Feb 02 2024
%e A173300 a(3) = 2 because x^3 + y^3 = 5/2.
%p A173300 nmax:=45: f:=n-> coeftayl((1+x)/(1-2*x-11*x^2-6*x^3), x=-1, n): a(1):=1: for n from 0 to nmax do a(n+2):= denom(f(n)) od: seq(a(n),n=1..nmax); # _Johannes W. Meijer_, Aug 16 2010
%t A173300 Denominator[Map[First, NestList[{Last[#], Last[#] + First[#]/2} &, {1, 2}, 50]]] (* _Paolo Xausa_, Feb 01 2024, after _Nick Hobson_ *)
%o A173300 (PARI) a(n) = denominator(2*polcoeff( lift( Mod((1+x)/2,x^2-3)^n ), 0)) \\ _Max Alekseyev_, Feb 23 2010
%o A173300 (Python)
%o A173300 from fractions import Fraction
%o A173300 def a173300_gen(a, b):
%o A173300     while True:
%o A173300         yield a.denominator
%o A173300         b, a = b + Fraction(a, 2), b
%o A173300 for n, a_n in zip(range(1, 47), a173300_gen(1, 2)):
%o A173300     print(n, a_n)  # _Nick Hobson_, Jan 30 2024
%Y A173300 Cf. A173299, A179596.
%Y A173300 Cf. A173989 (2-adic valuations).
%K A173300 nonn,frac
%O A173300 1,3
%A A173300 _J. Lowell_, Feb 15 2010
%E A173300 More terms from _Max Alekseyev_, Feb 23 2010