login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

A family of triangles T(m), m >= 0, read by triangles and then by rows; let P(m) be the triangle made up of the first m+1 rows of Pascal's triangle (A007318), R(m) (respectively L(m)) be P(m) rotated to the right (respectively to the left), T(m)(n, k) = gcd(P(m)(n, k), R(m)(n, k), L(m)(n, k)).
2

%I #13 Sep 05 2024 08:09:09

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%T 1,1,1,1,1,2,1,1,3,3,1,1,2,3,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%U 6,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

%N A family of triangles T(m), m >= 0, read by triangles and then by rows; let P(m) be the triangle made up of the first m+1 rows of Pascal's triangle (A007318), R(m) (respectively L(m)) be P(m) rotated to the right (respectively to the left), T(m)(n, k) = gcd(P(m)(n, k), R(m)(n, k), L(m)(n, k)).

%C T(m) has bilateral symmetry and 3-fold rotational symmetry.

%H Rémy Sigrist, <a href="/A375873/a375873.png">Colored representation of T(1023)</a> (where the hue is function of T(1023)(n, k)).

%H Rémy Sigrist, <a href="/A375873/a375873_1.png">Colored representation of T(1023)</a> (where dark colors correspond to high values).

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>.

%F T(m)(n, k) = gcd(A007318(n, k), A007318(m-k, n-k), A007318(m-n+k, m-n)).

%e T(0) is:

%e 1

%e T(1) is:

%e 1

%e 1 1

%e T(2) is:

%e 1

%e 1 1

%e 1 1 1

%e T(3) is:

%e 1

%e 1 1

%e 1 2 1

%e 1 1 1 1

%e T(4) is:

%e 1

%e 1 1

%e 1 1 1

%e 1 1 1 1

%e 1 1 1 1 1

%e T(5) is:

%e 1

%e 1 1

%e 1 2 1

%e 1 3 3 1

%e 1 2 3 2 1

%e 1 1 1 1 1 1

%o (PARI) T(m, n, k) = gcd([binomial(n, k), binomial(m-k, n-k), binomial(m-n+k, m-n)])

%Y Cf. A007318, A375963.

%K nonn,tabf,easy

%O 0,15

%A _Rémy Sigrist_, Sep 01 2024