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

1,2

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).

Links

Table of n, a(n) for n=1..88.

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.

Example

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;
  ...

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

Crossrefs

Cf. A000045, A140253, A290258.

Sequence in context: A286060 A080704 A316094 * A244779 A162158 A018552

Adjacent sequences: A290256 A290257 A290258 * A290260 A290261 A290262

Keyword

nonn,base,tabf

Author

Emeric Deutsch, Sep 12 2017

Status

approved