A115356 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

Offset: 0

Author

Paul Barry, Jan 21 2006

Keywords

• easy
• nonn
• tabl

Comments

As a flat sequence, with starting offset=1, a(n) = 1 if n is a triangular number (A000217) or twice a square (A001105), otherwise 0. - Antti Karttunen, Jan 19 2025

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;
(row and column numbering added by _Antti Karttunen_, Jan 19 2025)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..101474; the first 450 rows of the triangle
• Wikipedia, Riordan array
• Index entries for characteristic functions.

Crossrefs

Row sums are A000034. Diagonal sums are A115357. Inverse is A115358.

Cf. A000217, A001105, A010054, A379480.

Cf. also A115359.

Programs

PARI

A115356off1(n) = (ispolygonal(n,3) || (!(n%2) && issquare(n/2))); \\ (This is one-based)
A115356(n) = A115356off1(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