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For any number k >= 0, let T_k be the triangle whose base corresponds to the ternary expansion of k (without leading zeros) and other values, say t above u and v, satisfy t = (-u-v) mod 3; this sequence lists the numbers k such that the configurations of 0's, 1's and 2's in T_k are the same up to rotation.
1

%I #12 Mar 28 2023 14:01:17

%S 3,5,6,7,11,15,19,21,84,93,102,140,149,158,168,177,186,196,205,214,

%T 308,318,351,377,410,420,528,532,574,588,702,715,2271,2396,2523,2621,

%U 2775,2873,2933,3150,3185,3375,3410,3627,3687,3785,3939,4037,4164,4289,4519

%N For any number k >= 0, let T_k be the triangle whose base corresponds to the ternary expansion of k (without leading zeros) and other values, say t above u and v, satisfy t = (-u-v) mod 3; this sequence lists the numbers k such that the configurations of 0's, 1's and 2's in T_k are the same up to rotation.

%C This sequence is a variant of A361818.

%C If k belongs to the sequence, then A004488(k) belongs to the sequence.

%C The ternary lengths of terms belong to A007494 (as the number of values in triangles must be divisible by 3).

%C This sequence is infinite as it contains the numbers whose ternary digits match the regular expression "(210)+".

%C Empirically, there are 4*3^floor((w-1)/2) terms with w ternary digits.

%C No term belongs to A297250.

%H Rémy Sigrist, <a href="/A361827/a361827.png">Triangles illustrating initial terms</a>

%H Rémy Sigrist, <a href="/A361827/a361827.gp.txt">PARI program</a>

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

%e The ternary expansion of 149 is "12112", and the corresponding triangle is:

%e 0

%e 1 2

%e 0 2 2

%e 0 0 1 0

%e 1 2 1 1 2

%e The configurations of 0's, 1's and 2's are the same up to rotation, so 149 belongs to this sequence:

%e 0 . .

%e . . 1 . . 2

%e 0 . . . . . . 2 2

%e 0 0 . 0 . . 1 . . . . .

%e . . . . . 1 . 1 1 . . 2 . . 2

%o (PARI) See Links section.

%Y Cf. A004488, A007494, A297250, A361818.

%K nonn,base

%O 1,1

%A _Rémy Sigrist_, Mar 26 2023