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

%I A117567 #18 Jan 19 2025 14:16:27
%S A117567 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 A117567 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 A117567 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 A117567 Riordan array ((1+x^2)/(1-x^3),x).
%C A117567 Sequence array for the sequence F(L((n+2)/3)).
%D A117567 Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.
%H A117567 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 A117567 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 A117567 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.
%F A117567 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 A117567 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 A117567 Triangle begins:
%e A117567 n\k|  0  1  2  3  4  5  6  7  8  9
%e A117567 ---+--------------------------------
%e A117567 0  |  1,
%e A117567 1  |  0, 1,
%e A117567 2  |  1, 0, 1,
%e A117567 3  |  1, 1, 0, 1,
%e A117567 4  |  0, 1, 1, 0, 1,
%e A117567 5  |  1, 0, 1, 1, 0, 1,
%e A117567 6  |  1, 1, 0, 1, 1, 0, 1,
%e A117567 7  |  0, 1, 1, 0, 1, 1, 0, 1,
%e A117567 8  |  1, 0, 1, 1, 0, 1, 1, 0, 1,
%e A117567 9  |  1, 1, 0, 1, 1, 0, 1, 1, 0, 1
%e A117567 etc. Row and column numbering added by _Antti Karttunen_, Jan 19 2025
%o A117567 (PARI)
%o A117567 up_to = 119;
%o A117567 A117567tr0(n,k) = abs(kronecker((n-k+2), 3)); \\ We could also use fibonacci instead of abs
%o A117567 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 A117567 v117567 = A117567list(up_to);
%o A117567 A117567(n) = v117567[1+n]; \\ _Antti Karttunen_, Jan 19 2025
%Y A117567 Row sums are A093878. Diagonal sums are A051275. Inverse is A117568.
%K A117567 easy,nonn,tabl
%O A117567 0,1
%A A117567 _Paul Barry_, Mar 29 2006
%E A117567 Data section extended up to a(119) [15 rows of triangle] by _Antti Karttunen_, Jan 19 2025