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A173300
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a(n) is the denominator of the fraction f = x^n + y^n given that x + y = 1 and x^2 + y^2 = 2.
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4
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1, 1, 2, 2, 4, 2, 8, 8, 16, 8, 32, 32, 64, 32, 128, 128, 256, 128, 512, 512, 1024, 512, 2048, 2048, 4096, 2048, 8192, 8192, 16384, 8192, 32768, 32768, 65536, 32768, 131072, 131072, 262144, 131072, 524288, 524288, 1048576, 524288, 2097152, 2097152, 4194304, 2097152
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = denominator of ((1+sqrt(3))/2)^n + ((1-sqrt(3))/2)^n. - Max Alekseyev, Feb 23 2010
Conjecture: a(n) = 4*a(n-4), for n >= 7. - Paolo Xausa, Feb 02 2024
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EXAMPLE
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a(3) = 2 because x^3 + y^3 = 5/2.
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MAPLE
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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
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MATHEMATICA
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Denominator[Map[First, NestList[{Last[#], Last[#] + First[#]/2} &, {1, 2}, 50]]] (* Paolo Xausa, Feb 01 2024, after Nick Hobson *)
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PROG
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(PARI) a(n) = denominator(2*polcoeff( lift( Mod((1+x)/2, x^2-3)^n ), 0)) \\ Max Alekseyev, Feb 23 2010
(Python)
from fractions import Fraction
def a173300_gen(a, b):
while True:
yield a.denominator
b, a = b + Fraction(a, 2), b
for n, a_n in zip(range(1, 47), a173300_gen(1, 2)):
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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