A036585 Ternary Thue-Morse sequence: closed under a->abc, b->ac, c->b.

3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

First differences of A001969. Observed by Franklin T. Adams-Watters, proved by Max Alekseyev, Aug 30 2006

References

• M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 26.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001969, A007413, A005679.

Programs

Haskell

a036585 n = a036585_list !! (n-1)
a036585_list = 3 : concat (map f a036585_list)
where f 1 = [1,2,3]; f 2 = [1,3]; f 3 = [2]
-- Reinhard Zumkeller, Oct 31 2012

Mathematica

Differences[ThueMorse[Range[0, 100]]] + 2 (* Paolo Xausa, Oct 25 2024 *)

PARI

a(n)=if(n<1 || valuation(n,2)%2,2,2-(-1)^subst(Pol(binary(n)),x,1))

Python

def A036585(n): return 2+(n.bit_count()&1)-((n-1).bit_count()&1) # Chai Wah Wu, Mar 03 2023

Formula

a(n) = A001969(n+1) - A001969(n). - Franklin T. Adams-Watters, Aug 28 2006

a(n) = A029883(n) + 2 = A036577(n) + 1.