%I #9 Dec 09 2021 01:00:39

%S 0,0,1,0,0,1,0,1,0,1,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,1,

%T 0,1,0,0,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,

%U 1,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Triangle T(n,k) = 1 if both n and k are even or if n and k are divisible by 3.

%C Excludes S(n,k) such that gcd(n,k) = 1.

%C Row n in {1,5} mod 6 consists of k zeros; column k in {1,5} mod 6 is always 0.

%C Row or column p > 5 where p is prime consists of p zeros.

%C For n = 0 (mod 6), k in A047229 have T(n,k) = 1.

%C For k = 0 (mod 6), n in A047229 have T(n,k) = 1.

%C T(n,k) such that n and k both belong to {2,3,4} mod 6 form a "quincunx" or x-shaped checkerboard pattern evident in the table. In A054521, these have the value 0 along with other terms T(n,k) such that gcd(n,k) > 1.

%H Michael De Vlieger, <a href="/A349297/b349297.txt">Table of n, a(n) for n = 1..11325</a> (rows 1 <= n <= 150, flattened)

%H Michael De Vlieger, <a href="/A349297/a349297.png">Bitmap</a> magnified 3X showing T(n,k) = 1 in black and 0 in white.

%e Table T(n,k) for 1 <= n <= 16, replacing 0 with "." to accentuate the pattern:

%e 1: .

%e 2: . 1

%e 3: . . 1

%e 4: . 1 . 1

%e 5: . . . . .

%e 6: . 1 1 1 . 1

%e 7: . . . . . . .

%e 8: . 1 . 1 . 1 . 1

%e 9: . . 1 . . 1 . . 1

%e 10: . 1 . 1 . 1 . 1 . 1

%e 11: . . . . . . . . . . .

%e 12: . 1 1 1 . 1 . 1 1 1 . 1

%e 13: . . . . . . . . . . . . .

%e 14: . 1 . 1 . 1 . 1 . 1 . 1 . 1

%e 15: . . 1 . . 1 . . 1 . . 1 . . 1

%e 16: . 1 . 1 . 1 . 1 . 1 . 1 . 1 . 1

%e ---------------------------------------------------

%e n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

%t Table[Array[Boole@ Or[Mod[#, 2] == Mod[n, 2] == 0, Mod[#, 3] == Mod[n, 3] == 0] &, n], {n, 13}]

%Y Cf. A005843, A008585, A047229, A054521.

%K nonn,easy,tabl

%O 1

%A _Michael De Vlieger_, Nov 13 2021