A293670 - OEIS

A293670

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

%I #9 Nov 11 2019 00:50:28

%S 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,

%T 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,

%U 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

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

%C Numeric characterization:

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

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

%C - Iteration n consists of

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

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

%C 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:

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

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

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

%H Luc Rousseau, <a href="/A293670/a293670_1.pdf">Relation between A293670 and A002024 - Numeric structure of an anamorphosis</a>

%e Array begins (characterization)(x stands for -1):

%e 1 x

%e 0 2

%e 1 0 2 x

%e 1 2 0 3

%e 1 1 2 0 3 x

%e 2 2 1 3 0 4

%e 1 2 2 1 3 0 4 x

%e 3 2 2 3 1 4 0 5

%e 1 3 2 2 3 1 4 0 5 x

%e 4 2 3 3 2 4 1 5 0 6

%e 1 4 2 3 3 2 4 1 5 0 6 x

%e 5 2 4 3 3 4 2 5 1 6 0 7

%e 1 5 2 4 3 3 4 2 5 1 6 0 7 x

%e Or (definition)(to be read by antidiagonals):

%e x x x x

%e 1 2 2 3 3 4 4 5 ...

%e 0 0 0 0

%e 1 2 2 3 3 4 4 5 ...

%e 1 1 1 1

%e 1 2 2 3 3 4 4 5 ...

%e 2 2 2 2

%e 1 2 2 3 3 4 4 5 ...

%e 3 3 3 3

%e 1 2 2 3 3 4 4 5 ...

%e 4 4 4 4

%e ...

%o (PARI)

%o evolve(L,n)=if(n%2==1,listinsert(L,1,1);listinsert(L,-1,#L+1),L=apply(v->n/2-v,L));L

%o N=30;L=List();for(n=1,N,L=evolve(L,n);for(i=1,#L,print1(L[i],", "));print())

%Y Cf. A293578, A002024.

%K sign,tabf

%O 1,4

%A _Luc Rousseau_, Oct 14 2017