OFFSET
1,4
COMMENTS
The Duchêne et al. (2011) reference mentions many other sequences that are of great interest.
From Kevin Ryde, Dec 26 2020: (Start)
This sequence can be taken as a square array S(m,k) read by upwards antidiagonals with rows m >= 0 and columns k >= 0. S(m,k) is the final state for input word a^m b^k where ^ denotes repetition. Input a's cycle around states 0,1,2,3 so S(m,0) = m mod 4. Within an array row, input b's are no change at state 0 (so a row of 0's), or a repeating cycle 3,2,1 starting at m mod 4.
Antidiagonal d of the array is input words of length d = m+k, so terms S(d-k,k). These are words a^(d-k) b^k and the combination of d-k mod 4 and k mod 3 is 12-periodic within a diagonal. The sequence can also be taken as a triangle read by rows T(d,k) = S(d-k,k) for d >= 0 and 0 <= k <= d.
Rigo (2000, section 2.1 remark 2) notes that the sequence (flat sequence) is not periodic because pairs of terms 0,0 are at ever-increasing distances apart. They are a(n)=a(n+1)=0 iff n = 2*t*(4*t+1) = A033585(t) for t >= 1, which is every fourth triangular number.
(End)
LINKS
Eric Duchêne, Aviezri S. Fraenkel, Richard J. Nowakowski, and Michel Rigo, Extensions and restrictions of Wythoff's game preserving Wythoff's sequence as set of P-positions, Slides from a talk, LIAFA, Paris, October 21, 2011. See around the 35th slide, a slide with first line "In fact, this is a special case of the following result...".
Michel Rigo, Generalization of automatic sequences for numeration systems on a regular language, Theoret. Comput. Sci., 244 (2000) 271-281. See section 2.1 sequence u.
Michel Rigo and Arnaud Maes, More on generalized automatic sequences, Journal of Automata, Languages and Combinatorics 7.3 (2002): 351-376. See Fig. 3.
FORMULA
From Kevin Ryde, Dec 26 2020: (Start)
S(m,k) = 0 if m==0 (mod 4), otherwise S(m,k) = (((m mod 4) - k - 1) mod 3) + 1.
T(d,k) = S(d-k,k) = p(3*d+k mod 12) where p(0..11) = 0,2,3,1, 0,1,2,3, 0,3,1,2.
(End)
EXAMPLE
From Kevin Ryde, Dec 26 2020: (Start)
Array S(m,k) begins
k=0 1 2 3 4 5 6 7
+-------------------------
m=0 | 0, 0, 0, 0, 0, 0, 0, 0,
m=1 | 1, 3, 2, 1, 3, 2, 1,
m=2 | 2, 1, 3, 2, 1, 3, sequence by upwards
m=3 | 3, 2, 1, 3, 2, antidiagonals,
m=4 | 0, 0, 0, 0,
m=5 | 1, 3, 2, 12-periodic in diagonals
m=6 | 2, 1, (3 or 1-periodic in rows)
m=7 | 3, (4-periodic in columns)
(End)
PROG
(Python)
aut0, aut1 = [1, 2, 3, 0], [0, 3, 1, 2]
a, row = [0], [0]
for i in range(1, 10):
row = [aut0[row[0]]] + [aut1[x] for x in row]
a += row
print(a)
# Andrey Zabolotskiy, Aug 17 2018
(PARI) S(m, k) = if(m%=4, (m-k-1)%3+1, 0); \\ Kevin Ryde, Dec 26 2020
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Aug 12 2018
EXTENSIONS
New name and terms a(51) and beyond from Andrey Zabolotskiy, Aug 17 2018
STATUS
approved