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A134660 Number of odd coefficients in (1 + x + x^2 + x^3)^n.

%I A134660 #42 Jan 01 2018 04:19:07
%S A134660 1,4,4,4,4,16,4,8,4,16,16,4,4,16,8,16,4,16,16,16,16,64,4,8,4,16,16,8,
%T A134660 8,32,16,32,4,16,16,16,16,64,16,32,16,64,64,4,4,16,8,16,4,16,16,16,16,
%U A134660 64,8,16,8,32,32,16,16,64,32,64,4,16,16,16,16,64,16,32,16,64,64,16,16,64
%N A134660 Number of odd coefficients in (1 + x + x^2 + x^3)^n.
%H A134660 G. C. Greubel, <a href="/A134660/b134660.txt">Table of n, a(n) for n = 0..10000</a>
%H A134660 S. R. Finch, P. Sebah and Z.-Q. Bai, <a href="http://arXiv.org/abs/0802.2654">Odd Entries in Pascal's Trinomial Triangle</a>, arXiv:0802.2654 [math.NT], 2008.
%F A134660 a(n) = 2^A036555(n).
%F A134660 a(n) = gcd(4^n, C(4*n, n)). - _Peter Luschny_, Nov 08 2011
%e A134660 From _Omar E. Pol_, Mar 01 2015: (Start)
%e A134660 Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
%e A134660 1;
%e A134660 4;
%e A134660 4,4;
%e A134660 4,16,4,8;
%e A134660 4,16,16,4,4,16,8,16;
%e A134660 4,16,16,16,16,64,4,8,4,16,16,8,8,32,16,32;
%e A134660 4,16,16,16,16,64,16,32,16,64,64,4,4,16,8,16,4,16,16,16,16,64,8,16,8,32,32,16,16,64,32,64;
%e A134660 ...
%e A134660 (End)
%p A134660 seq(igcd(4^n,binomial(4*n,n)),n=0..77); # _Peter Luschny_, Nov 08 2011
%t A134660 PolynomialMod[(1+x+x^2+x^3)^n, 2] /. x->1
%t A134660 A036555 = Total /@ IntegerDigits[3 Range[0, 100], 2]; Table[2^A036555[[n]], {n, 1, 20}] (* or *) Table[GCD[4^n, Binomial[4*n, n]], {n, 0, 50}] (* _G. C. Greubel_, Dec 31 2017 *)
%o A134660 (PARI) a(n) = {my(pol= Pol([1,1,1,1], xx)*Mod(1,2)); subst(lift(pol^n), xx, 1);} \\ _Michel Marcus_, Mar 01 2015
%o A134660 (PARI) a(n) = 2^hammingweight(3*n); \\ _Joerg Arndt_, Mar 10 2015
%Y A134660 Cf. A000120, A001316, A008287, A036555, A071053.
%K A134660 nonn
%O A134660 0,2
%A A134660 _Steven Finch_, Jan 25 2008