A272636 a(0)=0, a(1)=1; thereafter a(n) = squarefree part of a(n-1)+a(n-2).

0, 1, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7

Offset

0,4

Comments

Periodic with period {1,2,3,5,2,7}.

James Propp, in a posting to the Math Fun list, asks if every sequence of positive numbers satisfying the same recurrence will eventually merge with this sequence (as A272638 does). The answer is no, Fred W. Helenius found infinitely many counterexamples, including A272637. See A272639 for other counterexamples which start 1,x.

Other counterexamples found by Helenius include [n, 2n, 3n, 5n, 2n, 7n] (period 6) where n is any squarefree positive integer coprime to 210 = 2*3*5*7.

Links

Table of n, a(n) for n=0..115.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 1).

Mathematica

{0, 1}~Join~LinearRecurrence[{0, 0, 0, 0, 0, 1}, {1, 2, 3, 5, 2, 7}, 120] (* Jean-François Alcover, Nov 16 2019 *)

Prog

(Python)
from sympy.ntheory.factor_ import core
l=[0, 1]
for n in range(2, 101):
    l.append(core(l[n - 1] + l[n - 2]))
print(l) # Indranil Ghosh, Jun 03 2017

Crossrefs

Cf. A007913 (squarefree part of n), A000045, A272637, A272638, A272639.

See A165911 for a similar sequence.

See also A214674, A214892-A214898.

Sequence in context: A053723 A201652 A359609 * A066949 A073481 A178094

Adjacent sequences: A272633 A272634 A272635 * A272637 A272638 A272639

Keyword

nonn

Author

N. J. A. Sloane, May 05 2016

Status

approved