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A174600 T(n,k) = 1 if the sum of +-k..+-n with arbitrary signs never equals zero, = 0 otherwise (lower triangle).

%I A174600 #21 Jul 09 2025 04:32:37
%S A174600 1,1,1,0,1,1,0,1,1,1,1,0,1,1,1,1,0,0,1,1,1,0,1,1,0,1,1,1,0,1,1,0,0,1,
%T A174600 1,1,1,0,0,1,1,0,1,1,1,1,0,0,1,1,1,0,1,1,1,0,1,1,0,0,1,1,0,1,1,1,0,1,
%U A174600 1,0,0,1,1,1,0,1,1,1,1,0,0,1,1,0,0,1,1,0,1,1,1,1,0,0,1,1,0,0,1,1,1,0,1,1,1,0,1,1
%N A174600 T(n,k) = 1 if the sum of +-k..+-n with arbitrary signs never equals zero, = 0 otherwise (lower triangle).
%H A174600 R. H. Hardin, <a href="/A174600/b174600.txt">Table of n, a(n) for n=1..4950</a>
%e A174600 Triangle begins
%e A174600 1;
%e A174600 1, 1;
%e A174600 0, 1, 1;
%e A174600 0, 1, 1, 1;
%e A174600 1, 0, 1, 1, 1;
%e A174600 1, 0, 0, 1, 1, 1;
%e A174600 0, 1, 1, 0, 1, 1, 1;
%e A174600 0, 1, 1, 0, 0, 1, 1, 1;
%e A174600 1, 0, 0, 1, 1, 0, 1, 1, 1;
%e A174600 1, 0, 0, 1, 1, 1, 0, 1, 1, 1; ...
%t A174600 t[n_, k_] := If[Mod[Floor[(n+1)/2], 2] != Mod[Floor[k/2], 2], 1, If[Mod[n-k, 2] == 0, If[k > ((n-k)/2)^2, 1, 0], 0]]; Flatten[Table[t[n, k], {n, 1, 15}, {k, 1, n}]][[;; 108]] (* _Jean-François Alcover_, Jul 11 2011, after awk program *)
%o A174600 (AWK)
%o A174600 { for(n=1; n<10; n++)
%o A174600         for(k=1; k<=n; k++)
%o A174600             print ++i, T(n,k);
%o A174600 }
%o A174600 function T(n,k) {
%o A174600     if ( int((n+1)/2)%2 != int(k/2)%2 ) return 1;
%o A174600     else if ( (n-k)%2 == 0 ) {
%o A174600         if ( k > ((n-k)/2)^2 ) return 1;
%o A174600         else return 0;
%o A174600     }
%o A174600     else return 0;
%o A174600 }
%o A174600 (PARI)
%o A174600 T(n,k)=
%o A174600 {
%o A174600     if ( ((n+1)\2)%2 != (k\2)%2,
%o A174600         return(1);
%o A174600     , /* else */
%o A174600         if ( (n-k)%2 == 0,
%o A174600             if ( k > ((n-k)/2)^2, return(1), return(0) );
%o A174600         , /* else */
%o A174600             return(0);
%o A174600         );
%o A174600     );
%o A174600 }
%o A174600 { for(n=1, 10, /* show triangle */
%o A174600     for(k=1,n,
%o A174600         print1(T(n,k),", ");
%o A174600     );
%o A174600     print();
%o A174600  ); }
%Y A174600 Related to A063865.
%K A174600 nonn,tabl
%O A174600 1,1
%A A174600 _R. H. Hardin_ Mar 23 2010, with formula and reference from Franklin T. Adams-Watters and _Olivier Gérard_, on the Sequence Fans Mailing list