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Period 8: repeat [0,1,0,1,1,0,1,0].
5

%I #54 Aug 30 2024 15:01:54

%S 0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,

%T 0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0,1,

%U 1,0,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0

%N Period 8: repeat [0,1,0,1,1,0,1,0].

%C Parity of A064706.

%C Parity of the generalized pentagonal numbers A001318. - _Omar E. Pol_, Feb 04 2012

%C More generally, parity of the generalized k-gonal numbers, for odd k >= 5. - _Omar E. Pol_, Feb 05 2012

%H Colin Barker, <a href="/A165211/b165211.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,-1,1).

%F a(n) = A002817(n) mod 2. - _Wesley Ivan Hurt_, Apr 23 2014

%F a(n) = 1/2 - (-1)^(n*(n+1)*(n^2 + n + 2)/8)/2. - _Vaclav Kotesovec_, Apr 28 2014

%F From _Colin Barker_, Dec 20 2017: (Start)

%F G.f.: x*(1 - x + x^2) / ((1 - x)*(1 + x^4)).

%F a(n) = a(n-1) - a(n-4) + a(n-5) for n>4.

%F (End)

%t PadRight[{},112,{0,1,0,1,1,0,1,0}] (* _Harvey P. Dale_, Jan 29 2012 *)

%t Table[Mod[n*(n+1)*(n^2+n+2)/8,2],{n,0,100}] (* _Vaclav Kotesovec_, Apr 28 2014 after _Wesley Ivan Hurt_ *)

%o (PARI) a(n)=bitxor(n, n\4)%2 \\ _Charles R Greathouse IV_, Jul 13 2016

%o (PARI) concat(0, Vec(x*(1 - x + x^2) / ((1 - x)*(1 + x^4)) + O(x^100))) \\ _Colin Barker_, Dec 20 2017

%o (Python)

%o def A165211(n): return n&1^bool(n&4) # _Chai Wah Wu_, Aug 30 2024

%Y Cf. A001318, A002817, A064706.

%Y Cf. A130198 (essentially the same).

%K nonn,easy

%O 0,1

%A _Philippe Deléham_, Sep 07 2009