A173300 - OEIS

%I #29 Feb 04 2024 18:39:09

%S 1,1,2,2,4,2,8,8,16,8,32,32,64,32,128,128,256,128,512,512,1024,512,

%T 2048,2048,4096,2048,8192,8192,16384,8192,32768,32768,65536,32768,

%U 131072,131072,262144,131072,524288,524288,1048576,524288,2097152,2097152,4194304,2097152

%N 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 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 a(n) = denominator of ((1+sqrt(3))/2)^n + ((1-sqrt(3))/2)^n. - _Max Alekseyev_, Feb 23 2010

%F Conjecture: a(n) = 4*a(n-4), for n >= 7. - _Paolo Xausa_, Feb 02 2024

%e a(3) = 2 because x^3 + y^3 = 5/2.

%p 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 Denominator[Map[First, NestList[{Last[#], Last[#] + First[#]/2} &, {1, 2}, 50]]] (* _Paolo Xausa_, Feb 01 2024, after _Nick Hobson_ *)

%o (PARI) a(n) = denominator(2*polcoeff( lift( Mod((1+x)/2,x^2-3)^n ), 0)) \\ _Max Alekseyev_, Feb 23 2010

%o (Python)

%o from fractions import Fraction

%o def a173300_gen(a, b):

%o     while True:

%o         yield a.denominator

%o         b, a = b + Fraction(a, 2), b

%o for n, a_n in zip(range(1, 47), a173300_gen(1, 2)):

%o     print(n, a_n)  # _Nick Hobson_, Jan 30 2024

%Y Cf. A173299, A179596.

%Y Cf. A173989 (2-adic valuations).

%K nonn,frac

%O 1,3

%A _J. Lowell_, Feb 15 2010

%E More terms from _Max Alekseyev_, Feb 23 2010