A070909 Triangle read by rows giving successive states of cellular automaton generated by "Rule 28" and by "Rule 156".

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

Offset: 0

Author

Hans Havermann, May 26 2002

Keywords

• nonn
• tabl

Comments

Row n has length n+1.

From Gary W. Adamson, May 15 2010: (Start)

Eigensequence of the triangle = A038754 (i.e., 1, 1, 2, 3, 6, 9, 18, ...) shifts to the left with multiplication by triangle A070909.

Binomial transform of A070909 = triangle A177953. (End)

From Paul Barry, Nov 03 2010: (Start)

Generalized (conditional) Riordan array with k-th column generated by x^k/(1-x) if k is even, x^k otherwise.

A181651 is an eigentriangle. Inverse is A181650. (End)

From Peter Bala, Aug 15 2021: (Start)

Double Riordan array (1/(1 - x); x*(1 - x), x/(1 - x)) as defined in Davenport et al. The inverse array is the double Riordan array (1 - x - x^2; x/(1 - x - x^2), x*(1 - x - x^2)).

In general, double Riordan arrays of the form (g(x); x/g(x), x*g(x)), where g(x) = 1 + g_1*x + g_2*x^2 + ..., form a group under matrix multiplication with the group law given by (g(x); x/g(x), x*g(x)) * (G(x); x/G(x), x*G(x)) = (h(x); x/h(x), x*h(x)), where h(x) = G(x) + (g(x) - 1)*(G(x) + G(-x))/2. The inverse array of (g(x); x/g(x), x*g(x)) equals (f(x); x/f(x), x*f(x)), where f(x) = (2 - (g(x) - g(-x)))/(g(x) + g(-x)). (End)

Examples

			From _Paul Barry_, Nov 03 2010: (Start)
Triangle begins
  1;
  1, 1;
  1, 0, 1;
  1, 0, 1, 1;
  1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 1;
  1, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1, 1;
  1, 0, 1, 0, 1, 0, 1, 0, 1;
  1, 0, 1, 0, 1, 0, 1, 0, 1, 1;
Production matrix begins
  1,  1;
  0, -1,  1;
  0, -1,  1,  1;
  0,  0,  0, -1,  1;
  0,  0,  0, -1,  1,  1;
  0,  0,  0,  0,  0, -1,  1;
  0,  0,  0,  0,  0, -1,  1,  1;
  0,  0,  0,  0,  0,  0,  0, -1,  1;
  0,  0,  0,  0,  0,  0,  0, -1,  1,  1;
  0,  0,  0,  0,  0,  0,  0,  0,  0, -1,  1; (End)
		

References

• S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Links

• Peter Bala, Matrices with repeated columns - the generalised Appell groups
• D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).
• Eric Weisstein's World of Mathematics, Rule 28
• Index entries for sequences related to cellular automata

Crossrefs

Inverse array A181650. Cf. A038754, A266502, A266508.

Programs

Mathematica

rows = 14; ca = CellularAutomaton[28, {{1}, 0}, rows-1]; Flatten[Table[ca[[k, 1 ;; k]], {k, 1, rows}]] (* Jean-François Alcover, May 24 2012 *)