A380123 Irregular table T(n, k), n >= 0, k = 1..A380122(n), read by rows; the n-th row lists the integers m (possibly negative) such that the nonzero digits in the nonadjacent form for m appear in the nonadjacent form for n.

0, 0, 1, 0, 2, -1, 0, 3, 4, 0, 4, 0, 1, 4, 5, -2, 0, 6, 8, -1, 0, 7, 8, 0, 8, 0, 1, 8, 9, 0, 2, 8, 10, -5, -4, -1, 0, 11, 12, 15, 16, -4, 0, 12, 16, -4, -3, 0, 1, 12, 13, 16, 17, -2, 0, 14, 16, -1, 0, 15, 16, 0, 16, 0, 1, 16, 17, 0, 2, 16, 18, -1, 0, 3, 4, 15, 16, 19, 20

Offset: 0

Author

Rémy Sigrist, Jan 12 2025

Keywords

• sign
• base
• tabf

Examples

			Irregular table T(n, k) begins:
  n   n-th row
  --  -----------------------------
   0  0
   1  0, 1
   2  0, 2
   3  -1, 0, 3, 4
   4  0, 4
   5  0, 1, 4, 5
   6  -2, 0, 6, 8
   7  -1, 0, 7, 8
   8  0, 8
   9  0, 1, 8, 9
  10  0, 2, 8, 10
  11  -5, -4, -1, 0, 11, 12, 15, 16
  12  -4, 0, 12, 16
  13  -4, -3, 0, 1, 12, 13, 16, 17
  14  -2, 0, 14, 16
  15  -1, 0, 15, 16
  16  0, 16
.
Irregular table T(n, k) begins in nonadjacent form:
  n      n-th row
  -----  ------------------------------------------
      0  0
      1  0, 1
     10  0, 10
    10T  T, 0, 10T, 100
    100  0, 100
    101  0, 1, 100, 101
   10T0  T0, 0, 10T0, 1000
   100T  T, 0, 100T, 1000
   1000  0, 1000
   1001  0, 1, 1000, 1001
   1010  0, 10, 1000, 1010
  10T0T  T0T, T00, T, 0, 10T0T, 10T00, 1000T, 10000
  10T00  T00, 0, 10T00, 10000
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..6724 (rows for n = 0..2^9 flattened)
• Joerg Arndt, Matters Computational (The Fxtbook), pages 61-62.
• Wikipedia, Non-adjacent form

Crossrefs

See A295989 for a similar sequence.

Cf. A184616, A380122.

Programs

PARI

row(n) = { my (r = [0], d, b = 1); while (n, if (n % 2, n -= d = 2 - (n%4); r = concat(r, [v + d*b | v <- r]);); n /= 2; b *= 2;); vecsort(r); }

Formula

T(n, 1) = A184616(n).

T(n, A380122(n)) = n.