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a(0)=0, a(1)=1; thereafter a(n) = squarefree part of a(n-1)+a(n-2).
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%I #40 Mar 03 2024 22:24:50

%S 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,

%T 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,

%U 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

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

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

%C _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.

%C 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.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 1).

%t {0, 1}~Join~LinearRecurrence[{0, 0, 0, 0, 0, 1}, {1, 2, 3, 5, 2, 7}, 120] (* _Jean-François Alcover_, Nov 16 2019 *)

%o (Python)

%o from sympy.ntheory.factor_ import core

%o l=[0, 1]

%o for n in range(2, 101):

%o l.append(core(l[n - 1] + l[n - 2]))

%o print(l) # _Indranil Ghosh_, Jun 03 2017

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

%Y See A165911 for a similar sequence.

%Y See also A214674, A214892-A214898.

%K nonn

%O 0,4

%A _N. J. A. Sloane_, May 05 2016