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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). 13
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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.

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

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

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

J.-P. Allouche, F. v. Haeseler, H.-O. Peitgen, A. Petersen and G. Skordev, Automaticity of double sequences generated by one-dimensional linear cellular automata, Theoret. Comput. Sci. 186 (1997), 195-209.

J.-P. Allouche, F. v. Haeseler, H.-O. Peitgen and G. Skordev, Linear cellular automata, finite automata and Pascal's triangle, Discrete Appl. Math. 66 (1996), 1-22.

R. Bacher, C. Reutenauer, The number of right ideals of given codimension over a finite field, in Noncommutative Birational Geometry, Representations and Combinatorics, edited by Arkady. Berenstein and Vladimir. Retakha, Contemporary Mathematics, Vol. 592, 2013.

Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, 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.

Index entries for characteristic functions

Index entries for sequences computed with run length transform

FORMULA

G.f.: 1 + x*A(x)^3 = A(x) (Mod 2); a(n) = A001764(n) (Mod 2).

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

Recurrence: a(0) = 1, a(2n) = a(4n+1) = a(n), a(4n+3) = 0.

a(n-2) = A000256(n)(mod 2), for n>2. - John M. Campbell, Jul 08 2016

a(n) = A000621(n+1)(mod 2). - John M. Campbell, Jul 15 2016

a(n) = A000625(n)(mod 2). - John M. Campbell, Jul 15 2016

a(n) = A008966(A005940(1+n)). [Follows from the Run Length Transform interpretation, see also A277010.] - Antti Karttunen, Oct 15 2016

a(n) = abs(A132971(n)) = abs(A008683(A005940(1+n))). - Antti Karttunen, May 30 2017

MAPLE

f:= proc(n) local L, Lp;

  L:= convert(n, base, 2);

  Lp:= convert(3*n, base, 2);

  if has(L-Lp[1..nops(L)], 1) then 0 else 1 fi

end proc:

map(f, [$0..100]); # Robert Israel, Jul 12 2016

MATHEMATICA

Table[Mod[Binomial[3 n, n], 2], {n, 0, 120}] (* Michael De Vlieger, Jul 08 2016 *)

PROG

(MAGMA) [Binomial(3*n, n) mod 2: n in [0..100]]; // Vincenzo Librandi, Jul 09 2016

CROSSREFS

Cf. A001764 (ternary trees), A085358 (runs of zeros), A076662 (runs of ones), A003849 (infinite Fibonacci word), A085359 (A(x)^3).

Cf. also A003714, A005940, A008966, A063524, A227349, A277010.

Absolute values of A132971.

Sequence in context: A097806 A167374 A132971 * A011748 A145361 A130304

Adjacent sequences:  A085354 A085355 A085356 * A085358 A085359 A085360

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 25 2003

STATUS

approved

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Last modified February 21 21:41 EST 2018. Contains 299427 sequences. (Running on oeis4.)