A115359 Matrix (1,x)-(x,x^2) in Riordan array notation.

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

Offset: 0

Author

Paul Barry, Jan 21 2006

Keywords

• easy
• sign
• tabl

Examples

			Triangle begins:
n\k|  0   1   2   3   4   5   6   7   8   9
---+-----------------------------------------
0  |  1;
1  | -1,  1;
2  |  0,  0,  1;
3  |  0, -1,  0,  1;
4  |  0,  0,  0,  0,  1;
5  |  0,  0, -1,  0,  0,  1;
6  |  0,  0,  0,  0,  0,  0,  1;
7  |  0,  0,  0, -1,  0,  0,  0,  1;
8  |  0,  0,  0,  0,  0,  0,  0,  0,  1;
9  |  0,  0,  0,  0, -1,  0,  0,  0,  0,  1;
etc. Row and column numbering added by _Antti Karttunen_, Jan 19 2025
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..22154; the first 210 rows of the triangle

Crossrefs

Row sums are 1,0,1,0,1,0... (A059841), Diagonal sums are A115360. Inverse is A115361.

Cf. A000217, A001105, A010054, A379480.

Cf. also A115356.

Programs

PARI

tabl(nn) = {T = matrix(nn, nn, n, k, n--; k--; if ((n==k), 1, if (n==2*k+1, -1, 0))); for (n=1, nn, for (k=1, n, print1(T[n, k], ", ");); print(););} \\ Michel Marcus, Mar 28 2015

PARI

A115359off1(n) = (ispolygonal(n,3)-(!(n%2) && issquare(n/2))); \\ (This is one-based)
A115359(n) = A115359off1(1+n); \\ (zero-based) - Antti Karttunen, Jan 19 2025

Formula

Number triangle T(n, k)=if(n=k, 1, 0) OR if(n=2k+1, -1, 0).

a(n) = A010054(n) - A379480(n). [As a flat sequence with starting offset 1] - Antti Karttunen, Jan 19 2025