A361818 - OEIS

%I #15 Mar 28 2023 14:01:21

%S 0,1,2,4,8,13,26,34,40,46,59,65,80,112,121,130,224,233,242,304,364,

%T 424,518,578,728,772,862,925,1003,1093,1183,1261,1324,1414,1535,1598,

%U 1688,1766,1856,1919,2006,2096,2186,2257,2509,2734,3028,3280,3532,3826,4051

%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 T_k has 3-fold rotational symmetry.

%C We can devise a similar sequence for any fixed base b >= 2; the present sequence corresponds to b = 3, and A334556 corresponds to b = 2.

%C This sequence is infinite as it contains A048328.

%C If k belongs to the sequence, then A004488(k) and A030102(k) belong to the sequence.

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

%C For any k, if t appears above u and v in T_k, then t + u + v = 0 (mod 3) and #{t, u, v} = 1 or 3 (the three values are either equal or all distinct); each value is uniquely determined by the two others in the same way: t = (-u-v) mod 3, u = (-t-v) mod 3, v = (-t-u) mod 3; this means that we can reconstruct T_k from any of its three sides.

%C If some row of T_k, say r, has w values and corresponds to the ternary expansion of m, then the row above r corresponds to the w-1 rightmost digits of the ternary expansion of A060587(m).

%C All positive terms belong to A297250 (their most significant digit equals their least significant digit in base 3).

%H Rémy Sigrist, <a href="/A361818/a361818.png">Triangles illustrating initial terms</a>

%H Rémy Sigrist, <a href="/A361818/a361818.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 304 is "102021", and the corresponding triangle is:

%e              1

%e             0 2

%e            2 1 0

%e           0 1 1 2

%e          2 1 1 1 0

%e         1 0 2 0 2 1

%e As this triangle has 3-fold rotational symmetry, 304 belongs to the sequence.

%o (PARI) See Links section.

%Y Cf. A004488, A048328, A060587, A297250, A334556.

%K nonn,base

%O 1,3

%A _Rémy Sigrist_, Mar 25 2023