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For any number k >= 0, let T_k be the triangle with values in {-1, 0, +1} whose base corresponds to the balanced ternary expansion of k (without leading zeros) and other values, say t above u and v, satisfy t+u+v = 0 mod 3; this sequence lists the numbers k such that T_k has 3-fold rotational symmetry.
2

%I #15 Apr 03 2024 10:07:01

%S 0,1,4,13,19,25,40,103,112,121,154,214,364,442,505,595,673,763,826,

%T 913,1003,1093,1144,1369,1621,1915,2167,2392,2776,3028,3280,3628,4420,

%U 4996,5668,6244,7036,8203,9022,9841,10459,10594,11782,12304,13411,13627,14419

%N For any number k >= 0, let T_k be the triangle with values in {-1, 0, +1} whose base corresponds to the balanced ternary expansion of k (without leading zeros) and other values, say t above u and v, satisfy t+u+v = 0 mod 3; this sequence lists the numbers k such that T_k has 3-fold rotational symmetry.

%C This sequence is a variant of A334556 and A361818.

%C This sequence is infinite as it contains A003462.

%C Empirically, for any w > 0, there are A127975(w-1) terms with w balanced ternary digits (ignoring leading zeros).

%C If k is a term then A338246(k) is also a term.

%H Rémy Sigrist, <a href="/A371636/a371636.png">Triangles illustrating initial terms</a> (blue, gray and red respectively denote -1's, 0's and 1's)

%H Rémy Sigrist, <a href="/A371636/a371636.gp.txt">PARI program</a>

%H <a href="/index/X#XOR-triangles">Index entries for sequences related to XOR-triangles</a>

%e The balanced ternary expansion of 595 is "1T11001" (where T denotes -1), and the corresponding triangle T_595 is as follows:

%e 1

%e T 0

%e 1 0 0

%e 1 1 T 1

%e 0 T 0 1 1

%e 0 0 1 T 0 T

%e 1 T 1 1 0 0 1

%e As this triangle has 3-fold rotational symmetry, 595 belongs to the sequence.

%o (PARI) \\ See Links section.

%Y Cf. A003462, A127975, A334556, A338246, A361818, A371635.

%K nonn,base

%O 1,3

%A _Rémy Sigrist_, Mar 30 2024