A304685 a(n) = A000699(n) (mod 3).

1, 1, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset: 1

Author

John M. Campbell, May 16 2018

Keywords

• nonn

Comments

This is a 3-automatic sequence.

Examples

			We have that A000699(5) = 248, with 248 == 2 (mod 3), and the ternary expansion of 5 is given by the tuple (1, 2), so according to the above formula we have that a(5) = 2.
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..2000
• John M. Campbell, A DFA for enumerating even-order irreducible diagrams modulo 3
• Index entries for 3-automatic sequences

Crossrefs

Cf. A000699.

Programs

GAP

a:=[1];;  for n in [2..90] do a[n]:=(n-1)*Sum([1..n-1],i->a[i]*a[n-i]) mod 3; od; a; # Muniru A Asiru, Aug 15 2018

Maple

a:=proc(n) option remember; if n<=1 then 1; else
add((2*i-1)*procname(i)*procname(n-i),i=1..n-1) mod 3; end if; end proc:
seq(a(n), n=1..90); # Muniru A Asiru, Aug 15 2018

PARI

a(n) = {my(A); A = O(x) ; for( i=1, n, A = x + A * (2 * x * A' - A)); polcoeff(A, n) % 3}; \\ Michel Marcus, Jul 04 2018; after A000699

Formula

For a natural number n, we have that a(n) = 1 if the ternary expansion of n is of the form 100...0 or is of the form 11...1200...0 for an even number of ones in this latter case, allowing runs of integers to be of length 0; a(n) = 2 if the ternary expansion of n is of the form 11...1200...0 if the length of the run of ones is odd; otherwise, a(n) = 0.