%I #18 Jan 19 2025 14:16:27

%S 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,

%T 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,

%U 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

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

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

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

%H Antti Karttunen, <a href="/A117567/b117567.txt">Table of n, a(n) for n = 0..101474; the first 450 rows of the triangle</a>

%H D. Panario, M. Sahin, Q. Wang, <a href="http://www.emis.de/journals/INTEGERS/papers/n78/n78.Abstract.html">A family of Fibonacci-like conditional sequences</a>, INTEGERS, Vol. 13, 2013, #A78.

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.

%F 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.

%F 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

%e Triangle begins:

%e n\k| 0 1 2 3 4 5 6 7 8 9

%e ---+--------------------------------

%e 0 | 1,

%e 1 | 0, 1,

%e 2 | 1, 0, 1,

%e 3 | 1, 1, 0, 1,

%e 4 | 0, 1, 1, 0, 1,

%e 5 | 1, 0, 1, 1, 0, 1,

%e 6 | 1, 1, 0, 1, 1, 0, 1,

%e 7 | 0, 1, 1, 0, 1, 1, 0, 1,

%e 8 | 1, 0, 1, 1, 0, 1, 1, 0, 1,

%e 9 | 1, 1, 0, 1, 1, 0, 1, 1, 0, 1

%e etc. Row and column numbering added by _Antti Karttunen_, Jan 19 2025

%o (PARI)

%o up_to = 119;

%o A117567tr0(n,k) = abs(kronecker((n-k+2), 3)); \\ We could also use fibonacci instead of abs

%o 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); };

%o v117567 = A117567list(up_to);

%o A117567(n) = v117567[1+n]; \\ _Antti Karttunen_, Jan 19 2025

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

%K easy,nonn,tabl

%O 0,1

%A _Paul Barry_, Mar 29 2006

%E Data section extended up to a(119) [15 rows of triangle] by _Antti Karttunen_, Jan 19 2025