OFFSET
0,2
COMMENTS
FORMULA
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.
EXAMPLE
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,...}.
PROG
(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])
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jul 21 2004
STATUS
approved