A290258 Triangle read by rows: row n (>=2) contains in increasing order the integers for which the binary representation has length n and all runs of 1's have even length.

3, 6, 12, 15, 24, 27, 30, 48, 51, 54, 60, 63, 96, 99, 102, 108, 111, 120, 123, 126, 192, 195, 198, 204, 207, 216, 219, 222, 240, 243, 246, 252, 255, 384, 387, 390, 396, 399, 408, 411, 414, 432, 435, 438, 444, 447, 480, 483, 486, 492, 495, 504, 507, 510

Offset: 2

Author

Emeric Deutsch, Sep 12 2017

Keywords

• nonn
• tabf

Comments

The viabin numbers of integer partitions having only even 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 [6,4,4,2]. The southeast border of its Ferrers board yields 110110011 (length is 9), leading to the viabin number 435 (a term in row 9).

Number of entries in row n is the Fibonacci number F(n-1) = A000045(n-1).

T(n,k) = A290259(n-1,k) + 2^(n-1).

Last entry in row n = A141023(n).

Basically the same as A277335.

Examples

			399 is in the sequence because all the runs of 1's of its binary representation, namely 110001111, have even lengths.
Triangle begins:
  3;
  6;
  12,15;
  24,27,30;
  48,51,54,60,63;
  96,99,102,108,111,120,123,126;
  ...
		

Crossrefs

Cf. A000045, A141023, A277335, A290259.

Programs

Maple

A[2] := {3}; A[3] := {6}; for n from 4 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

A[2] = {3}; A[3] = {6};
For[n = 4, n <= 10, n++, A[n] = Union[2 A[n-1], 4 A[n-2] + 3]];
Table[A[n], {n, 2, 10}] // Flatten (* Jean-François Alcover, Aug 19 2024, after Maple program *)

Formula

The entries in row n (n>=4) 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.