A350774 - OEIS

%I #7 Jan 16 2022 11:08:04

%S 2,3,4,5,4,5,6,7,6,7,8,9,8,9,10,11,10,9,8,9,8,9,10,11,10,11,12,13,12,

%T 13,14,15,14,13,12,13,12,13,14,15,14,15,16,17,16,17,18,19,18,17,16,17,

%U 16,17,18,19,18,19,20,21,20,21,22,23,22,21,20,21,20,19

%N Abelian complexity of the Rudin-Shapiro sequence (A020985).

%C a(n) = the number of contiguous length-n blocks in A020985, where two blocks that differ only the order of their letters are considered the same.

%H X. Lü, J. Chen, Z. Wen, and W. Wu, <a href="https://doi.org/10.1016/j.jmaa.2017.02.019">On the abelian complexity of the Rudin-Shapiro sequence</a>, J. Math. Anal. Appl. 451 (2017), 822-838.

%F a(4n+1) = 2a(n)

%F a(4n+3) = -2a(n) + 2a(4n+2)

%F a(8n) = -2a(n) + 2a(2n) + a(4n)

%F a(8n+2) = a(2n) + a(2n+1)

%F a(16n+4) = 4a(n) - a(2n+1) + a(8n+4)

%F a(16n+6) = 2a(n) + a(4n+2)

%F a(16n+12) = -4a(n) - 2a(2n+1) + 4a(4n+2) + a(8n+4)

%F a(16n+14) = -4a(n) - 2a(2n+1) + 4a(4n+2) + a(8n+4)

%e For n=5 there are 24 distinct subwords, but only 4 up to abelian equivalence.

%Y Cf. A020985.

%K nonn

%O 1,1

%A _Jeffrey Shallit_, Jan 15 2022