login
Natural numbers mapped to Dyck path encodings of the rooted plane trees obtained by recursing on the exponents of the GF(2)[X] factorization of n.
9

%I #10 Jul 11 2015 10:24:15

%S 0,10,1010,1100,110010,101100,101010,110100,10110010,11001100,

%T 10101010,10110100,1010101010,10101100,11010010,111000,11100010,

%U 1011001100,101010101010,1100110100,11001010,1010101100,101010110010

%N Natural numbers mapped to Dyck path encodings of the rooted plane trees obtained by recursing on the exponents of the GF(2)[X] factorization of n.

%C Note that we recurse on the exponent + 1 for all other irreducible polynomials except the largest one in the GF(2)[X] factorization. Thus for 6 = A048723(3,1) X A048723(2,1) we construct a tree by joining trees 1 and 2 with a new root node, for 7 = A048723(7,1) X A048723(3,0) X A048723(2,0) we join three 1-trees (single leaves) with a new root node, for 8 = A048273(2,3) we add a single edge below tree 3 and for 9 = A048723(7,1) X A048723(3,1) X A048273(2,0) we connect the trees 1 and 2 and 1 with a new root node.

%H A. Karttunen, <a href="/A091247/a091247.scm.txt">Scheme-program for computing this sequence.</a>

%e The rooted plane trees encoded here are:

%e .....................o....o..........o.........o...o....o.....

%e .....................|....|..........|..........\./.....|.....

%e .......o....o...o....o....o...o..o...o..o.o.o....o....o.o.o...

%e .......|.....\./.....|.....\./....\./....\|/.....|.....\|/....

%e *......*......*......*......*......*......*......*......*.....

%e 1......2......3......4......5......6......7......8......9.....

%Y a(n) = A007088(A106455(n)) = A075166(A106443(n)). GF(2)[X]-analog of A075166. Permutation of A063171. Same sequence shown in decimal: A106455. The digital length of each term / 2 (the number of o-nodes in the corresponding trees) is given by A106457. Cf. A106451-A106454.

%K nonn,base

%O 1,2

%A _Antti Karttunen_, May 09 2005