A204418 Periodic sequence 1,0,1,..., arranged in a triangle.

1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1

Offset: 0

Author

Philippe Deléham, Jan 15 2012

Keywords

• nonn
• tabl
• easy

Comments

Binomial transform is A130781.

Row sums: 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 9, ... = A004396(n+1) = A131737 (n+2) .

Diagonal sums: 1, 0, 2, 1, 1, 3, 3, 1, 5, 3, 2, 6, 5, 2, 8, 5, 3, 9, 7, 3, 11, 7, 4, 12, 9, 4, 14, 9, 5, 15, ..

Essentially the same as A141571 and A011655. - R. J. Mathar, Jan 16 2012

As sequence a(n) this is the characteristic sequence for the mod m reduced odd numbers (i.e., gcd(2*n+1,m)=1, n >= 0) for each modulus m from 3*A003586 = [3,6,9,12,18,24,27,36,48,...]. - Wolfdieter Lang, Feb 04 2012

Disregarding the triangle: a(A173732(n)) = 1. - Reinhard Zumkeller, Apr 29 2012

Examples

			Triangle begins:
  1;
  0, 1;
  1, 0, 1;
  1, 0, 1, 1;
  0, 1, 1, 0, 1;
  1, 0, 1, 1, 0, 1;
  1, 0, 1, 1, 0, 1, 1;
  0, 1, 1, 0, 1, 1, 0, 1;
  1, 0, 1, 1, 0, 1, 1, 0, 1;
		

References

• Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Crossrefs

Cf. A011655.

Programs

PARI

a(n)=n%3!=1 \\ Charles R Greathouse IV, Jul 13 2016

Formula

If k==0 mod(3), T(n+k,k) = 1, 0, 1, 1, 0, 1, 1, 0, 1, ... (A204418)

If k==1 mod(3), T(n+k,k) = 1, 0, 0, 1, 0, 0, 1, 0, 0, ... (A079978)

If n==2 mod(3), T(n+k,k) = 1, 1, 1, 1, 1, 1, 1, 1, 1, ... (A000012)

a(A016777(n)) = 0.

G.f.:(1+x^2)/(1-x^3).

G.f.: U(0) where U(k)= 1 + x^2/(1 - x/(x + 1/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 17 2012