A194045 Numbers whose binary expansion is a preorder traversal of a binary tree.

0, 4, 20, 24, 84, 88, 100, 104, 112, 340, 344, 356, 360, 368, 404, 408, 420, 424, 432, 452, 456, 464, 480, 1364, 1368, 1380, 1384, 1392, 1428, 1432, 1444, 1448, 1456, 1476, 1480, 1488, 1504, 1620, 1624, 1636, 1640, 1648, 1684, 1688, 1700, 1704, 1712, 1732, 1736, 1744, 1760, 1812, 1816, 1828, 1832, 1840, 1860, 1864, 1872, 1888, 1924, 1928, 1936, 1952, 1984

Offset: 0

Author

Mike Stay, Aug 13 2011

Keywords

• nonn

Comments

When traversing the tree in preorder, emit 1 at each node and 0 if it has no child on the current branch. The smallest binary tree is empty, so we emit 0 at the root; thus a(0) = 0. The next binary tree is a single node (emit 1) with no left (emit 0) or right child (emit 0); thus a(1) = 100 in binary or 4.

Crossrefs

Cf. A000108.

Programs

JavaScript

// n is the number of internal nodes (or 1s in the binary expansion)
// f is a function to display each result
function trees(n, f) {
  // h is the "height", thinking of 1 as a step up and 0 as a step down
  // s is the current state
  function enumerate(n, h, s, f) {
    if (n===0 && h===0) { f(2 * s); }
    else {
      if (h > 0) { enumerate(n, h - 1, 2 * s, f) }
      if (n > 0) { enumerate(n - 1, h + 1, 2 * s + 1, f) }
    }
  }
  enumerate(n, 0, 0, f);
}

Formula

a(n) = 4 * A057520(n). [Joerg Arndt, Sep 22 2012]

a(0)=0, a(n) = 2 * A014486(n) for n>=1. [Joerg Arndt, Sep 22 2012]