A295989 -id:A295989 - cp's OEIS frontend

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.

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.

A341691 a(0) = 0, and for any n > 0, a(n) = n - a(k) where k is the greatest number < n such that n AND a(k) = a(k) (where AND denotes the bitwise AND operator).

Original entry on oeis.org

0, 1, 2, 1, 4, 1, 2, 5, 8, 1, 2, 9, 4, 9, 10, 5, 16, 1, 2, 17, 4, 17, 18, 5, 8, 17, 18, 9, 20, 9, 10, 21, 32, 1, 2, 33, 4, 33, 34, 5, 8, 33, 34, 9, 36, 9, 10, 37, 16, 33, 34, 17, 36, 17, 18, 37, 40, 17, 18, 41, 20, 41, 42, 21, 64, 1, 2, 65, 4, 65, 66, 5, 8, 65

Offset: 0

Rémy Sigrist, Feb 17 2021

This sequence is a binary variant of A341679; here we look for a term whose binary 1's match those of n, there we look for a term that divides n.

			The first terms, alongside the corresponding value of k, are:
  n   a(n)  k
  --  ----  ---
   0     0  N/A
   1     1    0
   2     2    0
   3     1    2
   4     4    0
   5     1    4
   6     2    4
   7     5    6
   8     8    0
   9     1    8

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Rémy Sigrist, C program for A341691

Cf. A006519, A295989, A341679.

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a(n) = n iff n = 0 or n is a power of 2.
a(2*n) = 2*a(n).
Apparently, a(n) = n - a(n - A006519(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.

A374448 Irregular table T(n, k), n >= 0, 0 <= k < A089898(n), read by rows; the n-th row lists the numbers m in the range 0..n such that m and n-m can be added without carries in base 10.

Original entry on oeis.org

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

Offset: 0

Rémy Sigrist, Jul 08 2024

The n-th row lists the numbers m such that for any k > 0, the k-th rightmost digit of m is <= the k-th rightmost digit of n.

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

Rémy Sigrist, Table of n, a(n) for n = 0..9077 (rows for n = 0..200 flattened)

Cf. A002262, A089898, A295989 (base-2 analog).

T(n, k, base = 10) = { my (v = 0, p = 1, d, t); while (n, d = n % base; n \= base; t = k % (d+1); k \= (d+1); v += t * p; p *= base;); return (v); }

T(n, 0) = 0.
T(n, A089898(n)-1) = n.
T(n, k) + T(n, A089898(n)-1-k) = n.
T(10*n, k) = 10*T(n, k).

cp's OEIS Frontend

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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A341691 a(0) = 0, and for any n > 0, a(n) = n - a(k) where k is the greatest number < n such that n AND a(k) = a(k) (where AND denotes the bitwise AND operator).

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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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A374448 Irregular table T(n, k), n >= 0, 0 <= k < A089898(n), read by rows; the n-th row lists the numbers m in the range 0..n such that m and n-m can be added without carries in base 10.

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