OFFSET
0
COMMENTS
Conjecture: a(n) = A151666(floor((n+1)/2)) for n > 0. - Georg Fischer, Nov 29 2022
Conjecture confirmed. See the attached file. The idea is to guess the automaton for this sequence (in base 2) and then verify that it satisfies the identities in the Gawron-Ulas paper. Next we make a (base 2) automaton for A151666. This is easy because it is just the numbers consisting only of 0 and 1 in base 4. Then finally we assert Fischer's identity and Walnut returns TRUE. - Jeffrey Shallit, Nov 30 2022
LINKS
R. J. Mathar, Table of n, a(n) for n = 0..17000
Maciej Gawron and Maciej Ulas, On formal inverse of the Prouhet-Thue-Morse sequence, Discrete Mathematics 339.5 (2016): 1459-1470. Also arXiv:1601.04840 [math.CO], 2016.
Jeffrey Shallit, Verification of Fischer's conjecture
FORMULA
a(0)=0, a(1)=a(2)=1, a(3)=0; thereafter
if n mod 4 = 0 then a(n) = a(n-1),
if n mod 4 = 1 then a(n) = a(n-2),
if n mod 4 = 2 then a(n) = a(n-3),
otherwise a(n) = (a(n-4) + a((n-3)/4)) mod 2.
a(n) = A001002(n) mod 2 for n > 0. - John M. Campbell, Jul 17 2016
MAPLE
MATHEMATICA
a[n_] := a[n] = Which[n <= 3, {0, 1, 1, 0}[[n + 1]], Mod[n, 4] == 0, a[n - 1], Mod[n, 4] == 1, a[n - 2], Mod[n, 4] == 2, a[n - 3], True, Mod[a[n - 4] + a[(n - 3)/4], 2]];
Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Nov 27 2017, from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Apr 02 2016
STATUS
approved