A290259 Triangle read by rows: row n (>=1) contains in increasing order the integers for which the binary representation has length n, the first run of 1's has odd length, and all the other runs of 1's have even length.

1, 2, 4, 7, 8, 11, 14, 16, 19, 22, 28, 31, 32, 35, 38, 44, 47, 56, 59, 62, 64, 67, 70, 76, 79, 88, 91, 94, 112, 115, 118, 124, 127, 128, 131, 134, 140, 143, 152, 155, 158, 176, 179, 182, 188, 191, 224, 227, 230, 236, 239, 248, 251, 254, 256, 259, 262, 268, 271, 280, 283, 286, 304, 307, 310, 316, 319, 352, 355, 358, 364, 367, 376, 379, 382, 448, 451, 454, 460, 463, 472, 475, 478, 496, 499, 502, 508, 511

Offset: 1

Author

Emeric Deutsch, Sep 12 2017

Keywords

• nonn
• base
• tabf

Comments

The viabin numbers of integer partitions having only odd parts. The viabin number of an integer partition is defined in the following way. Consider the southeast border of the Ferrers board of the integer partition and consider the binary number obtained by replacing each east step with 1 and each north step, except the last one, with 0. The corresponding decimal form is, by definition, the viabin number of the given integer partition. "Viabin" is coined from "via binary". For example, consider the integer partition [7,5]. The southeast border of its Ferrers board yields 11111011 (length is 8), leading to the viabin number 251 (a term in row 8).

Number of entries in row n is the Fibonacci number F(n) = A000045(n).

T(n,k) = A290258(n+1,k) - 2^n.

Last entry in row n = A140253(n).

Examples

			115 is in the sequence; indeed, its binary representation, namely 1110011, has first run of 1's of odd length and the other runs of 1's have even length.
Triangle begins:
   1;
   2;
   4,  7;
   8, 11, 14;
  16, 19, 22, 28, 31;
  32, 35, 38, 44, 47, 56, 59, 62;
  ...
		

Crossrefs

Cf. A000045, A140253, A290258.

Programs

Maple

A[1] := {1}; A[2] := {2}; for n from 3 to 10 do A[n] := `union`(map(proc (x) 2*x end proc, A[n-1]), map(proc (x) 4*x+3 end proc, A[n-2])) end do; # yields sequence in triangular form

Mathematica

nmax = 10; A[1] = {1}; A[2] = {2}; For[n = 3, n <= nmax, n++, A[n] = Union[2 A[n-1], 4 A[n-2] + 3]]; Table[A[n], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Aug 26 2024, after Maple program *)

Formula

The entries in row n (n>=3) are (i) 2x, where x is in row n-1, and (ii) 4y + 3, where y is in row n-2. The Maple program is based on this.