A353174 -id:A353174 - cp's OEIS frontend

A356097 A family of triangles T(m), m >= 0, read by triangles and then by rows; triangle T(0) is [1; 1, 1]; for m >= 0, triangle T(m+1) is obtained by replacing each subtriangle [t; u, v] in T(m) by [t; t, t; u, t+u+v, v; u, u, v, v].

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 3, 3, 5, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 5, 3, 3, 5, 1, 1, 1, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 3, 3, 5, 1

Offset: 0

Rémy Sigrist, Jul 26 2022

We apply the following substitutions to transform T(m) into T(m+1):
                            t
                           / \
                          /   \
       t                 t-----t
      / \    _\       / \   / \
     /   \      /      /   \ /   \
    u-----v           u---t+u+v---v
                     / \   / \   / \
                    /   \ /   \ /   \
                   u-----u-----v-----v
and:
                   u-----u-----v-----v
                    \   / \   / \   /
                     \ /   \ /   \ /
    u-----v           u---t+u+v---v
     \   /   _\      \   / \   /
      \ /       /       \ /   \ /
       t                 t-----t
                          \   /
                           \ /
                            t
T(m) has 3^m+1 rows.
All terms are odd.
As m gets larger, T(m) exhibits interesting fractal features (see illustrations in Links section).

			Triangle T(0) is:
              1
             1 1
Triangle T(1) is:
              1
             1 1
            1 3 1
           1 1 1 1
Triangle T(2) is:
              1
             1 1
            1 3 1
           1 1 1 1
          1 1 5 1 1
         1 5 3 3 5 1
        1 1 3 3 3 1 1
       1 1 5 3 3 5 1 1
      1 3 1 1 5 1 1 3 1
     1 1 1 1 1 1 1 1 1 1

Rémy Sigrist, Colored representation of T(6) (the color is function of T(6)(n,k))
Rémy Sigrist, Representation of the multiples of 3 in T(7)
Rémy Sigrist, Representation of the multiples of 5 in T(7)
Rémy Sigrist, Representation of the multiples of 7 in T(7)
Rémy Sigrist, Representation of the 1's in T(7)
Rémy Sigrist, Representation of the terms congruent to 1 mod 4 in T(7)
Rémy Sigrist, PARI program
Rémy Sigrist, Nonperiodic tilings related to Stern's diatomic series and based on tiles decorated with elements of Fp, arXiv:2301.06039 [math.CO], 2023.
Wikipedia, Hexaflake

See A355855, A356002, A356096 and A356098 for similar sequences.

Cf. A353174.

PARI
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See Links section.
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A352502 a(n) is the number of integers k in the interval 0..n such that k and n-k can be added without carries in balanced ternary.

Original entry on oeis.org

1, 2, 2, 4, 4, 2, 4, 4, 6, 10, 8, 6, 10, 8, 2, 4, 4, 6, 10, 8, 6, 10, 8, 10, 16, 12, 18, 28, 20, 14, 22, 16, 10, 16, 12, 18, 28, 20, 14, 22, 16, 2, 4, 4, 6, 10, 8, 6, 10, 8, 10, 16, 12, 18, 28, 20, 14, 22, 16, 10, 16, 12, 18, 28, 20, 14, 22, 16, 18, 28, 20, 30

Offset: 0

Rémy Sigrist, Apr 28 2022

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 Gould's sequence (A001316); here we work with balanced ternary, there with binary.

			For n = 8:
- we consider the following cases:
              k|    0    1    2    3    4    5    6    7    8
      ---------+---------------------------------------------
        bter(k)|    0    1   1T   10   11  1TT  1T0  1T1  10T
      bter(8-k)|  10T  1T1  1T0  1TT   11   10   1T    1    0
       carries?|  no   yes  no   no   yes  no   no   yes  no
- so a(8) = 6.

Rémy Sigrist, Table of n, a(n) for n = 0..6561
Wikipedia, Balanced ternary

Cf. A001316, A059095, A140429, A353174 (corresponding k's).

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) }
a(n) = sum(k=0, n, ok(n-k, k))

a(n) <= n+1 with equality iff n belongs to A140429.
a(3*n) = 3*a(n) - 2.
a(3*n+1) = a(3*n-1) + 2.

A380273 Irregular table T(n, k), n >= 0, k = 1..A380272(n), read by rows; the n-th row lists the integers m in 0..n such that the nonadjacent forms for m-n and m can be added without carries.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 3, 4, 0, 1, 4, 5, 0, 6, 0, 7, 0, 1, 2, 6, 7, 8, 0, 1, 8, 9, 0, 2, 8, 10, 0, 11, 0, 1, 11, 12, 0, 1, 12, 13, 0, 14, 0, 3, 4, 11, 12, 15, 0, 1, 2, 3, 4, 5, 11, 12, 13, 14, 15, 16, 0, 1, 4, 5, 12, 13, 16, 17, 0, 2, 16, 18, 0, 3, 4, 15, 16, 19

Offset: 0

Rémy Sigrist, Jan 18 2025

The nonadjacent forms for two integers, say Sum_{i >= 0} x_i * 2^i and Sum_{i >= 0} y_i * 2^i, can be added without carries iff for any i >= 0:
- abs(x_i + y_i) <= 1,
- (x_i + y_i) * (x_{i+1} + y_{i+1}) = 0.

			Table T(n, k) begins:
  n   n-th row
  --  ----------------------------------------
   0  0
   1  0, 1
   2  0, 2
   3  0, 3
   4  0, 1, 3, 4
   5  0, 1, 4, 5
   6  0, 6
   7  0, 7
   8  0, 1, 2, 6, 7, 8
   9  0, 1, 8, 9
  10  0, 2, 8, 10
  11  0, 11
  12  0, 1, 11, 12
  13  0, 1, 12, 13
  14  0, 14
  15  0, 3, 4, 11, 12, 15
  16  0, 1, 2, 3, 4, 5, 11, 12, 13, 14, 15, 16

Rémy Sigrist, Table of n, a(n) for n = 0..10524
Joerg Arndt, Matters Computational (The Fxtbook), pages 61-62.
Rémy Sigrist, Scatterplot of (n, T(n, k)) for n = 0..2^9 (in a hexagonal lattice)
Wikipedia, Non-adjacent form

See A295989 and A353174 for similar sequences.

Cf. A380272.

ok(x, y) = { my (dx, dy, p = 0, q); while (x || y, if (x % 2, x -= dx = 2 - (x%4), dx = 0); if (y % 2, y -= dy = 2 - (y%4), dy = 0); if (dx && dx==dy, return (0);); q = dx + dy; if (p && q, return (0);); x /= 2; y /= 2; p = q;); return (1); }
row(n) = select(k -> ok(n-k, k), [0..n])

T(n, 1) = 0.

T(n, A380272(n)) = n.

A363930 Irregular table T(n, k), n >= 0, k = 1..A363710(n), read by rows; the n-th row lists the nonnegative numbers m <= n such that A003188(m) AND A003188(n-m) = 0 (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 3, 4, 0, 1, 4, 5, 0, 6, 0, 7, 0, 1, 7, 8, 0, 1, 2, 3, 6, 7, 8, 9, 0, 2, 3, 7, 8, 10, 0, 3, 8, 11, 0, 1, 3, 9, 11, 12, 0, 1, 12, 13, 0, 14, 0, 15, 0, 1, 15, 16, 0, 1, 2, 3, 14, 15, 16, 17, 0, 2, 3, 4, 6, 12, 14, 15, 16, 18, 0, 3, 4, 7, 12, 15, 16, 19

Offset: 0

Rémy Sigrist, Jun 28 2023

This sequence is related to the T-square fractal (see A363710).

			Table T(n, k) begins:
  n   n-th row
  --  ----------------------
   0  0
   1  0, 1
   2  0, 2
   3  0, 3
   4  0, 1, 3, 4
   5  0, 1, 4, 5
   6  0, 6
   7  0, 7
   8  0, 1, 7, 8
   9  0, 1, 2, 3, 6, 7, 8, 9
  10  0, 2, 3, 7, 8, 10
  11  0, 3, 8, 11
  12  0, 1, 3, 9, 11, 12
  13  0, 1, 12, 13
  14  0, 14
  15  0, 15
  16  0, 1, 15, 16

Rémy Sigrist, Table of n, a(n) for n = 0..13126 (rows for n = 0..2^9 flattened)
Wikipedia, T-square (fractal)

See A295989, A353174 and A362327 for similar sequences.

Cf. A003188, A363710.

row(n) = { select (m -> bitand(bitxor(m, m\2), bitxor(n-m, (n-m)\2))==0, [0..n]) }

T(n, 1) = 0.
T(n, A363710(n)) = n.
T(n, k) + T(n, A363710(n)+1-k) = n.

cp's OEIS Frontend

A356097 A family of triangles T(m), m >= 0, read by triangles and then by rows; triangle T(0) is [1; 1, 1]; for m >= 0, triangle T(m+1) is obtained by replacing each subtriangle [t; u, v] in T(m) by [t; t, t; u, t+u+v, v; u, u, v, v].

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A352502 a(n) is the number of integers k in the interval 0..n such that k and n-k can be added without carries in balanced ternary.

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A380273 Irregular table T(n, k), n >= 0, k = 1..A380272(n), read by rows; the n-th row lists the integers m in 0..n such that the nonadjacent forms for m-n and m can be added without carries.

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A363930 Irregular table T(n, k), n >= 0, k = 1..A363710(n), read by rows; the n-th row lists the nonnegative numbers m <= n such that A003188(m) AND A003188(n-m) = 0 (where AND denotes the bitwise AND operator).

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