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 A293670 Square array made of (W, N, S, E) quadruplets read by antidiagonals. Numeric structure of an anamorphosis of A002024 (see comments). 1
 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 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A293578, A002024. Sequence in context: A140082 A330714 A343640 * A279792 A025852 A190762 Adjacent sequences:  A293667 A293668 A293669 * A293671 A293672 A293673 KEYWORD sign,tabf AUTHOR Luc Rousseau, Oct 14 2017 STATUS approved

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Last modified January 19 11:42 EST 2022. Contains 350465 sequences. (Running on oeis4.)