A293670 Square array made of (W, N, S, E) quadruplets read by antidiagonals. Numeric structure of an anamorphosis of A002024 (see comments).

1, -1, 0, 2, 1, 0, 2, -1, 1, 2, 0, 3, 1, 1, 2, 0, 3, -1, 2, 2, 1, 3, 0, 4, 1, 2, 2, 1, 3, 0, 4, -1, 3, 2, 2, 3, 1, 4, 0, 5, 1, 3, 2, 2, 3, 1, 4, 0, 5, -1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, -1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, -1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, -1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, -1

Offset: 1

Author

Luc Rousseau, Oct 14 2017

Keywords

• sign
• tabf

Comments

Numeric characterization:

Row n is the value of a list after n iterations of the following algorithm:

- start with an empty list (assimilable to row number 0)

- Iteration n consists of

-- if n is odd, appending 1 to the left of the list and -1 to the right;

-- if n is even, replacing each value in the list by its complement to n/2.

Underlying definition and interest: this sequence represents a square array in which each cell is a structure made of 4 values arranged in W/N/S/E fashion. These values are twice the areas of elementary right triangles that enter the composition of quadrilaterals delimited by two families of lines, with the following equations:

- for m = 1, 2, 3, ...: y = mx - (m-1)^2 {x <= m-1}

- for n = -1, 0, 1, ...: y = -nx - (n+1)^2 {x >= 1-n}

Globally these quadrilaterals form an anamorphosis of A002024. See provided link for explanations and illustrations.

Examples

			Array begins (characterization)(x stands for -1):
              1 x
              0 2
            1 0 2 x
            1 2 0 3
          1 1 2 0 3 x
          2 2 1 3 0 4
        1 2 2 1 3 0 4 x
        3 2 2 3 1 4 0 5
      1 3 2 2 3 1 4 0 5 x
      4 2 3 3 2 4 1 5 0 6
    1 4 2 3 3 2 4 1 5 0 6 x
    5 2 4 3 3 4 2 5 1 6 0 7
  1 5 2 4 3 3 4 2 5 1 6 0 7 x
Or (definition)(to be read by antidiagonals):
    x     x     x     x
  1   2 2   3 3   4 4   5 ...
    0     0     0     0
    0     0     0     0
  1   2 2   3 3   4 4   5 ...
    1     1     1     1
    1     1     1     1
  1   2 2   3 3   4 4   5 ...
    2     2     2     2
    2     2     2     2
  1   2 2   3 3   4 4   5 ...
    3     3     3     3
    3     3     3     3
  1   2 2   3 3   4 4   5 ...
    4     4     4     4
  ...
		

Links

• Luc Rousseau, Relation between A293670 and A002024 - Numeric structure of an anamorphosis

Crossrefs

Cf. A293578, A002024.

Programs

PARI

evolve(L,n)=if(n%2==1,listinsert(L,1,1);listinsert(L,-1,#L+1),L=apply(v->n/2-v,L));L
N=30;L=List();for(n=1,N,L=evolve(L,n);for(i=1,#L,print1(L[i],", "));print())