A077606 Left differencing matrix, D, by antidiagonals.

1, -1, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1,1

Comments

If v is a sequence written as a column vector, then Dv is the sequence of first differences of v. The inverse of D is the left summing matrix; the transpose of D is the right differencing matrix.

Links

Table of n, a(n) for n=1..108.

C. Kimberling, Matrix Transformations of Integer Sequences, J. Integer Seqs., Vol. 6, 2003.

Formula

D(n, n-1)=-1, D(n, n)=1, else D(n, k)=0.

As a sequence, a(2k^2-2k+1) = 1, a(2k^2) = -1, otherwise a(n) = 0. - Franklin T. Adams-Watters, Jan 12 2007

Example

Northwest corner:
1 0 0 0 0
-1 1 0 0 0
0 -1 1 0 0
0 0 -1 1 0
0 0 0 -1 1

Crossrefs

Cf. A077605.

Cf. A001844, A001105.

Sequence in context: A113429 A133100 A216230 * A004601 A372552 A074937

Adjacent sequences: A077603 A077604 A077605 * A077607 A077608 A077609

Keyword

easy,sign,tabl

Author

Clark Kimberling, Nov 11 2002

Status

approved