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A085357 Common residues of binomial(3n,n)/(2n+1) modulo 2: relates ternary trees (A001764) to the infinite Fibonacci word (A003849).

%I A085357 #65 Jun 25 2025 15:11:32
%S A085357 1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,1,
%T A085357 1,0,1,1,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,
%U A085357 1,1,0,0,1,1,1,0,0,0,0,0,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A085357 Common residues of binomial(3n,n)/(2n+1) modulo 2: relates ternary trees (A001764) to the infinite Fibonacci word (A003849).
%C A085357 The n-th runs of ones is given by: 3 - A003849(n) (infinite Fibonacci word) = A076662(n+1). Runs of zeros are given by: A085358 and are also directly related to the Fibonacci sequence. Coefficients of A(x)^3 are found in A085359.
%C A085357 a(n) = 0 iff some binary digit of n is 1 while the corresponding binary digit of 3*n is 0. - _Robert Israel_, Jul 12 2016
%C A085357 The Run Length Transform of [0,1,0,0,0,...], A063524, the characteristic function of 1. (See A227349 for the definition). - _Antti Karttunen_, Oct 15 2016
%H A085357 Robert Israel, <a href="/A085357/b085357.txt">Table of n, a(n) for n = 0..10000</a>
%H A085357 J.-P. Allouche, F. v. Haeseler, H.-O. Peitgen, A. Petersen and G. Skordev, <a href="http://dx.doi.org/10.1016/S0304-3975(96)00298-8">Automaticity of double sequences generated by one-dimensional linear cellular automata</a>, Theoret. Comput. Sci. 186 (1997), 195-209.
%H A085357 J.-P. Allouche, F. v. Haeseler, H.-O. Peitgen and G. Skordev, <a href="http://dx.doi.org/10.1016/0166-218X(94)00132-W">Linear cellular automata, finite automata and Pascal's triangle</a>, Discrete Appl. Math. 66 (1996), 1-22.
%H A085357 R. Bacher and C. Reutenauer, <a href="http://bookstore.ams.org/conm-592/12">The number of right ideals of given codimension over a finite field</a>, in Noncommutative Birational Geometry, Representations and Combinatorics, edited by Arkady. Berenstein and Vladimir. Retakha, Contemporary Mathematics, Vol. 592, 2013.
%H A085357 Paul Tarau, <a href="http://dx.doi.org/10.1007/978-3-642-23283-1_15">Emulating Primality with Multiset Representations of Natural Numbers</a>, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238, DOI: 10.1007/978-3-642-23283-1_15.
%H A085357 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%H A085357 <a href="/index/Ru#rlt">Index entries for sequences computed with run length transform</a>
%F A085357 G.f.: 1 + x*A(x)^3 = A(x) (Mod 2); a(n) = A001764(n) (Mod 2).
%F A085357 a(n) = binomial(3n, n) (mod 2). Characteristic function of Fibbinary numbers (i.e. a(n)=1 iff n is in A003714). - _Benoit Cloitre_, Nov 15 2003
%F A085357 Recurrence: a(0) = 1, a(2n) = a(4n+1) = a(n), a(4n+3) = 0.
%F A085357 a(n-2) = A000256(n)(mod 2), for n>2. - _John M. Campbell_, Jul 08 2016
%F A085357 a(n) = A000621(n+1)(mod 2). - _John M. Campbell_, Jul 15 2016
%F A085357 a(n) = A000625(n)(mod 2). - _John M. Campbell_, Jul 15 2016
%F A085357 a(n) = A008966(A005940(1+n)). [Follows from the Run Length Transform interpretation, see also A277010.] - _Antti Karttunen_, Oct 15 2016
%F A085357 a(n) = abs(A132971(n)) = abs(A008683(A005940(1+n))). - _Antti Karttunen_, May 30 2017
%p A085357 f:= proc(n) local L,Lp;
%p A085357   L:= convert(n,base,2);
%p A085357   Lp:= convert(3*n,base,2);
%p A085357   if has(L-Lp[1..nops(L)],1) then 0 else 1 fi
%p A085357 end proc:
%p A085357 map(f, [$0..100]); # _Robert Israel_, Jul 12 2016
%t A085357 Table[Mod[Binomial[3 n, n], 2], {n, 0, 120}] (* _Michael De Vlieger_, Jul 08 2016 *)
%o A085357 (Magma) [Binomial(3*n,n) mod 2: n in [0..100]]; // _Vincenzo Librandi_, Jul 09 2016
%o A085357 (PARI) A085357(n) = !bitand(n,n<<1); \\ _Antti Karttunen_, Aug 22 2019
%o A085357 (Python)
%o A085357 def A085357(n): return int(not n&(n<<1)) # _Chai Wah Wu_, Jun 25 2025
%Y A085357 Cf. A001764 (ternary trees), A085358 (runs of zeros), A076662 (runs of ones), A003849 (infinite Fibonacci word), A085359 (A(x)^3).
%Y A085357 Cf. also A003714, A005940, A008966, A063524, A227349, A277010, A324964.
%Y A085357 Absolute values of A132971.
%K A085357 nonn
%O A085357 0,1
%A A085357 _Paul D. Hanna_, Jun 25 2003