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A078971 Numbers n such that C(4n,n)/(3n+1) (A002293) is not divisible by 4.

%I A078971 #27 Sep 08 2022 08:45:08
%S A078971 0,1,3,5,11,13,21,43,45,53,85,171,173,181,213,341,683,685,693,725,853,
%T A078971 1365,2731,2733,2741,2773,2901,3413,5461,10923,10925,10933,10965,
%U A078971 11093,11605,13653,21845,43691,43693,43701,43733,43861,44373,46421,54613
%N A078971 Numbers n such that C(4n,n)/(3n+1) (A002293) is not divisible by 4.
%C A078971 Stanica observes that the sequence in binary forms a pattern where 1 bits are inserted into the word 1010101...:
%C A078971 1 11
%C A078971 101 1011 1101
%C A078971 10101 101011 101101 110101
%C A078971 1010101 10101011 10101101 10110101 11010101...
%H A078971 Chai Wah Wu, <a href="/A078971/b078971.txt">Table of n, a(n) for n = 1..5050</a>
%H A078971 P. Stanica, <a href="http://arXiv.org/abs/math.NT/0010148">p^q Catalan numbers and squarefree binomial coefficients</a>, arXiv:math/0010148 [math.NT], 2000.
%t A078971 Select[ Range[0, 65000], Mod[ Binomial[4#, # ]/(3# + 1), 4] != 0 &] (* _Robert G. Wilson v_, Oct 12 2005 *)
%o A078971 (PARI) isok(n) = binomial(4*n,n)/(3*n+1) % 4; \\ _Michel Marcus_, Apr 16 2015
%o A078971 (Magma) [n: n in [0..2*10^4] | not IsZero(Binomial(4*n,n) div (3*n+1) mod 4)]; // _Vincenzo Librandi_, Apr 16 2015
%o A078971 (Python)
%o A078971 from __future__ import division
%o A078971 A078971_list = []
%o A078971 for t in range(100):
%o A078971     A078971_list.append((2**(2*t)-1)//3)
%o A078971     for j in range(t):
%o A078971         A078971_list.append((2**(2*t+1)+2**(2*j+1)-1)//3) # _Chai Wah Wu_, Mar 06 2016
%Y A078971 Cf. A000225 (C(2n, n)/(n+1) is not divisible by 2), A003462 (C(3n, n)/(2n+1) is not divisible by 3), A003463 (C(5n, n)/(4n+1) is not divisible by 5).
%K A078971 nonn
%O A078971 1,3
%A A078971 _Benoit Cloitre_, Jan 14 2003
%E A078971 Comments and more terms from _Ralf Stephan_, Oct 30 2003
%E A078971 a(28)-a(44) from _Robert G. Wilson v_, Oct 12 2005