0,2

Antidiagonal sums form A007494 (numbers congruent to 0 or 2 mod 3). Sums of squares of antidiagonals form A096808 and is congruent to A004526 mod 4. Using these terms as powers of (-1) results in the table of signs A096809.

For n>=0: T(0, n)=0, T(n+1, 0)=2, T(n+1, n+1)=1. T(k, n) = 3 - T(n, k) for n>0, k>=0, and n != k. Construction: start with T(0, 0)=0 and proceed for all i>=0 in this way: for k=0..2^i-1, concatenate the (2^i)x(2^i) matrix to itself to form a matrix twice its size: T(n, k+2^i)=T(n, k), T(n+2^i, k)=T(n, k), T(n+2^i, k+2^i)=T(n, k); then for n=0..2^i-1, increment these elements by +1: T(2^i, n), T(n+2^i, n), T(n+2^i, 2^i). Example: start with the matrix: 0 0 2 1 concatenate this matrix to itself to form a matrix twice the size: 0 0 | 0 0 2 1 | 2 1 ----+---- 0 0 | 0 0 2 1 | 2 1 then increment the elements that comprise the far left column of the matrix in the lower right quadrant, and those elements that comprise the top row and diagonal of the matrix in the lower left quadrant (the element found in both the top row and diagonal gets incremented twice): 0 0 | 0 0 2 1 | 2 1 ----+---- 2 1 | 1 0 2 2 | 3 1 Repeating these steps forms this table.

The elements in the table begin:

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1

2 1 1 0 2 1 1 0 2 1 1 0 2 1 1 0

2 2 3 1 2 2 3 1 2 2 3 1 2 2 3 1

2 1 1 1 1 0 0 0 2 1 1 1 1 0 0 0

2 2 2 1 3 1 2 1 2 2 2 1 3 1 2 1

2 1 2 0 3 1 1 0 2 1 2 0 3 1 1 0

2 2 3 2 3 2 3 1 2 2 3 2 3 2 3 1

2 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0

2 2 2 1 2 1 2 1 3 1 2 1 2 1 2 1

2 1 2 0 2 1 1 0 3 1 1 0 2 1 1 0

2 2 3 2 2 2 3 1 3 2 3 1 2 2 3 1

2 1 1 1 2 0 0 0 3 1 1 1 1 0 0 0

2 2 2 1 3 2 2 1 3 2 2 1 3 1 2 1

2 1 2 0 3 1 2 0 3 1 2 0 3 1 1 0

2 2 3 2 3 2 3 2 3 2 3 2 3 2 3 1

The sum of the antidiagonals begin: {0,2,3,5,6,8,9,11,12,14,...}.

(PARI) {T(n, k)=local(M, D=6); if(n<0|k<0, 0, M=matrix(2^D, 2^D); M[2, 1]=2; M[2, 2]=1; for(i=1, D-1, for(r=1, 2^i, for(c=1, 2^i, M[r, c+2^i]=M[r, c]; M[r+2^i, c]=M[r, c]; M[r+2^i, c+2^i]=M[r, c]); M[1+2^i, r]+=1; M[r+2^i, r]+=1; M[r+2^i, 1+2^i]+=1; )); M[n+1, k+1])}

Cf. A096808, A096809, A007494, A003987.

Adjacent sequences: A096807 A096808 A096809 this_sequence A096811 A096812 A096813

Sequence in context: A038760 A035394 A067167 this_sequence A133696 A127371 A036849

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Jul 21 2004