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A293670
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Square array made of (W, N, S, E) quadruplets read by antidiagonals. Numeric structure of an anamorphosis of A002024 (see comments).
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1
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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
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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
...
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PROG
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(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())
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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