OFFSET
1
COMMENTS
Sum of antidiagonals generates sequence A049820 beginning at n=3.
When written as a triangular array, TR(n,j), by shifting row n of square array to the right by n columns (i.e., TR(n,j) = T(n,k-n) see second part of Example below), the sum of antidiagonals then becomes the column sums. TR(n,j) is then the trivial indicator of all the proper factors of the integers (N) starting at N = 3, where a value of zero in the array indicates the presence of a proper factor. [More details on that if you like: the row index n+1 of a zero in any column of TR(n,j) is factor, N = j + 2 (column index + 2) is the integer of interest.]
FORMULA
T(n,k) = 1 if k mod (n+1)> 0; T(n,k) = 0 if k mod (n+1) = 0.
Or, in simple words, each row is a repeating pattern that starts with n instances of 1 followed by one instance of 0.
From Boris Putievskiy, May 08 2013: (Start)
As table T(n,k)= 1-floor(k/(n+1))+floor((k-1)/(n+1)).
a(n)=1-floor(j/(i+1))+floor((j-1)/(i+1)), where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). (End)
EXAMPLE
As the Square Array T(n,k)
1,0,1,0,1,0,1,0,1,0,1,0 ...
1,1,0,1,1,0,1,1,0,1,1,0 ...
1,1,1,0,1,1,1,0,1,1,1,0 ...
1,1,1,1,0,1,1,1,1,0,1,1 ...
1,1,1,1,1,0,1,1,1,1,1,0 ...
1,1,1,1,1,1,0,1,1,1,1,1 ...
Now, as a Triangular Array TR(n,j):
1,0,1,0,1,0,1,0,1,0,1,0 ...
0,1,1,0,1,1,0,1,1,0,1,1,0 ...
0,0,1,1,1,0,1,1,1,0,1,1,1,0 ...
0,0,0,1,1,1,1,0,1,1,1,1,0,1,1 ...
0,0,0,0,1,1,1,1,1,0,1,1,1,1,1,0 ...
0,0,0,0,0,1,1,1,1,1,1,0,1,1,1,1,1 ...
MATHEMATICA
max = 15; row[n_] := Table[{Table[1, {n}], 0}, {max/(n+1)}] // Flatten; t = Table[row[n], {n, max - 1}]; Table[t[[n-k+1, k]], {n, 1, max-1}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, May 02 2013 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Richard R. Forberg, May 01 2013
EXTENSIONS
More terms from Jean-François Alcover, May 02 2013
STATUS
approved