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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

Table of n, a(n) for n=1..162.

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

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.)