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

0,1

Links

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

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

Example

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

Prog

(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

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.

Sequence in context: A256538 A074910 A115356 * A117906 A090678 A110037

Adjacent sequences: A115356 A115357 A115358 * A115360 A115361 A115362

Keyword

easy,sign,tabl,changed

Author

Paul Barry, Jan 21 2006

Status

approved