OFFSET
1,4
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.
LINKS
EXAMPLE
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
...
PROG
(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())
CROSSREFS
KEYWORD
sign,tabf
AUTHOR
Luc Rousseau, Oct 14 2017
STATUS
approved