A117567 Riordan array ((1+x^2)/(1-x^3),x).

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

Offset: 0

Author

Paul Barry, Mar 29 2006

Keywords

• easy
• nonn
• tabl

Comments

Sequence array for the sequence F(L((n+2)/3)).

Examples

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

References

• Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..101474; the first 450 rows of the triangle
• D. Panario, M. Sahin, Q. Wang, A family of Fibonacci-like conditional sequences, INTEGERS, Vol. 13, 2013, #A78.
• Index entries for characteristic functions.

Crossrefs

Row sums are A093878. Diagonal sums are A051275. Inverse is A117568.

Programs

PARI

up_to = 119;
A117567tr0(n,k) = abs(kronecker((n-k+2), 3)); \\ We could also use fibonacci instead of abs
A117567list(up_to) = { my(v = vector(1+up_to), i=0); for(n=0,oo, for(k=0,n, i++; if(i > 1+up_to, return(v)); v[i] = A117567tr0(n,k))); (v); };
v117567 = A117567list(up_to);
A117567(n) = v117567[1+n]; \\ Antti Karttunen, Jan 19 2025

Formula

Number triangle T(n,k) = F(L((n-k+2)/3))[k<=n] where L(j/p) is the Legendre symbol of j and p.

In the above, I assume that F stands for Fibonacci sequence (A000045), which in domain {-1, 0, 1} reduces to taking the absolute value of the argument. - Antti Karttunen, Jan 19 2025

Extensions

Data section extended up to a(119) [15 rows of triangle] by Antti Karttunen, Jan 19 2025