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.

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

Links

Paolo Xausa, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,0,0,4).

Formula

a(n) = denominator of ((1+sqrt(3))/2)^n + ((1-sqrt(3))/2)^n = 2^max{0, n - A007814(A080040(n))}. - Max Alekseyev, Feb 23 2010

a(n) = 4*a(n-4), for n >= 7. - conjectured by Paolo Xausa, Feb 02 2024; proved by Max Alekseyev, Mar 16 2026

For n >= 3, a(n) = 2^((n-1)/2) if n == 1 (mod 2); 2^(n/2-1) if n == 0 (mod 4); 2^(n/2-2) if n == 2 (mod 4). - Max Alekseyev, Mar 16 2026

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 *)
LinearRecurrence[{0, 0, 0, 4}, {1, 1, 2, 2, 4, 2}, 50] (* Paolo Xausa, Mar 22 2026 *)

Prog

(PARI) a173300(n) = if(n==2, 1, 2^if(n%2==1, (n-1)/2, n/2-1-(n%4==2))); \\ Max Alekseyev, Mar 16 2026
(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. A173299, A179596.

Cf. A173989 (2-adic valuations).

Sequence in context: A303140 A370469 A103178 * A181236 A280684 A322671

Adjacent sequences: A173297 A173298 A173299 * A173301 A173302 A173303

Keyword

nonn,frac,easy

Author

J. Lowell, Feb 15 2010

Extensions

More terms from Max Alekseyev, Feb 23 2010

Status

approved