A361818 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.

0, 1, 2, 4, 8, 13, 26, 34, 40, 46, 59, 65, 80, 112, 121, 130, 224, 233, 242, 304, 364, 424, 518, 578, 728, 772, 862, 925, 1003, 1093, 1183, 1261, 1324, 1414, 1535, 1598, 1688, 1766, 1856, 1919, 2006, 2096, 2186, 2257, 2509, 2734, 3028, 3280, 3532, 3826, 4051

Offset: 1

Author

Rémy Sigrist, Mar 25 2023

Keywords

• nonn
• base

Comments

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.

This sequence is infinite as it contains A048328.

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

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

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.

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).

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

Examples

			The ternary expansion of 304 is "102021", and the corresponding triangle is:
             1
            0 2
           2 1 0
          0 1 1 2
         2 1 1 1 0
        1 0 2 0 2 1
As this triangle has 3-fold rotational symmetry, 304 belongs to the sequence.
		

Links

• Rémy Sigrist, Triangles illustrating initial terms
• Rémy Sigrist, PARI program
• Index entries for sequences related to XOR-triangles

Crossrefs

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

Programs

PARI

See Links section.