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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.
4
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
OFFSET
1,3
COMMENTS
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
FORMULA
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
EXAMPLE
a(3) = 2 because x^3 + y^3 = 5/2.
MAPLE
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
MATHEMATICA
Denominator[Map[First, NestList[{Last[#], Last[#] + First[#]/2} &, {1, 2}, 50]]] (* Paolo Xausa, Feb 01 2024, after Nick Hobson *)
PROG
(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)):
print(n, a_n) # Nick Hobson, Jan 30 2024
CROSSREFS
Cf. A173989 (2-adic valuations).
Sequence in context: A303140 A370469 A103178 * A181236 A280684 A322671
KEYWORD
nonn,frac
AUTHOR
J. Lowell, Feb 15 2010
EXTENSIONS
More terms from Max Alekseyev, Feb 23 2010
STATUS
approved