A353174 Irregular table T(n, k), n >= 0, k = 0..A352502(n)-1; the n-th row lists in ascending order the numbers k in 0..n such that k and n-k can be added without carries in balanced ternary.

0, 0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 3, 4, 0, 5, 0, 1, 5, 6, 0, 1, 6, 7, 0, 2, 3, 5, 6, 8, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 3, 4, 6, 7, 9, 10, 0, 2, 3, 8, 9, 11, 0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 0, 1, 3, 4, 9, 10, 12, 13, 0, 14, 0, 1, 14, 15, 0, 1, 15, 16, 0, 2, 3, 14, 15, 17

Offset: 0

Author

Rémy Sigrist, Apr 28 2022

Keywords

• nonn
• base
• tabf

Comments

Two integers can be added without carries in balanced ternary if they have no equal nonzero digit at the same position.

This sequence has connections with A295989; here we work with balanced ternary, there with binary.

The set of points {(n, T(n, k))} has interesting fractal features, with voids in the form of Koch snowflakes (see illustration in Links section).

Examples

			Irregular table T(n, k) begins:
    0: [0]
    1: [0, 1]
    2: [0, 2]
    3: [0, 1, 2, 3]
    4: [0, 1, 3, 4]
    5: [0, 5]
    6: [0, 1, 5, 6]
    7: [0, 1, 6, 7]
    8: [0, 2, 3, 5, 6, 8]
    9: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8768 (rows for n = 0..3^5 flattened)
• Rémy Sigrist, Scatterplot of (n, T(n, k)) for n <= 3^6 on a hexagonal lattice
• Wikipedia, Balanced ternary
• Wikipedia, Hexaflake.

Crossrefs

Cf. A059095, A295989, A352502.

Programs

PARI

ok(u,v) = { while (u && v, my (uu=[0,+1,-1][1+u%3], vv=[0,+1,-1][1+v%3]); if (abs(uu+vv)>1, return (0)); u=(u-uu)/3; v=(v-vv)/3); return (1) }
row(n) = select(k -> ok(n-k, k), [0..n])

Formula

T(n, 0) = 0.

T(n, A352502(n)-1) = n.

Extensions

Index correction from Rémy Sigrist, Jan 18 2025