A225145 Square array read by downwards antidiagonals: T(n,k) = 1 if k mod (n+1) > 0, T(n,k) = 0 if k mod (n+1) = 0.

1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1

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.]

Row n is characteristic function of numbers that are not multiples of n+1. Rows 1..9 are A059841, A011655, A166486, A011558, A097325, A109720, A168181, A168182, A168184. - Boris Putievskiy, May 08 2013

Links

Table of n, a(n) for n=1..105.

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)).

As linear sequence a(n) = 1-floor(A004736(n)/(A002260(n)+1) + floor((A004736(n)-1)/(A002260(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

Cf. A002260, A004736, A059841, A011655, A011558, A097325, A109720, A166486, A168181, A168182, A168184.

Sequence in context: A319102 A103582 A129565 * A121967 A356310 A285430

Adjacent sequences: A225142 A225143 A225144 * A225146 A225147 A225148

Keyword

nonn,tabl

Author

Richard R. Forberg, May 01 2013

Extensions

More terms from Jean-François Alcover, May 02 2013

Status

approved