A290251 a(n) is the number of parts in the integer partition having viabin number n.

0, 1, 2, 1, 3, 2, 2, 1, 4, 3, 3, 2, 3, 2, 2, 1, 5, 4, 4, 3, 4, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1, 6, 5, 5, 4, 5, 4, 4, 3, 5, 4, 4, 3, 4, 3, 3, 2, 5, 4, 4, 3, 4, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 1, 7, 6, 6, 5, 6, 5, 5, 4, 6, 5, 5, 4, 5, 4, 4, 3, 6, 5, 5, 4, 5, 4, 4, 3, 5, 4, 4, 3, 4, 3, 3, 2, 6, 5, 5, 4, 5, 4, 4, 3, 5, 4

Offset: 0

Author

Emeric Deutsch, Jul 24 2017

Keywords

• nonn
• tabf

Comments

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 [2,2,2,1]. The southeast border of its Ferrers board yields 10100, leading to the viabin number 20.

From Omar E. Pol, Jul 24 2017: (Start)

Consider that this is also an irregular triangle read by rows T(n,k), n>=0, k>=1, in which the row lengths are the elements of A011782 (see example).

Conjectures:

1) It appears that if k > 1 and k is a power of 2 then column k lists the positive integers (A000027).

2) It appears that column k lists the nonnegative integers (A001477) starting from the first elements of the column k.

3) It appears that if n > 0 then row n lists the first 2^(n-1) elements of A063787 in reverse order. (End)

Examples

			a(9) = 3. Indeed, the binary form of 9 is 1001; with an additional 0 at the end, it leads to the path ENNEN, where E=(1,0), N=(0,1); this path is the southeast border of the Ferrers board of the integer partition [2,1,1], having 3 parts.
From _Omar E. Pol_, Jul 24 2017: (Start)
Written as an irregular triangle the sequence begins:
  0;
  1;
  2,1;
  3,2,2,1;
  4,3,3,2,3,2,2,1;
  5,4,4,3,4,3,3,2,4,3,3,2,3,2,2,1;
  6,5,5,4,5,4,4,3,5,4,4,3,4,3,3,2,5,4,4,3,4,3,3,2,4,3,3,2,3,2,2,1;
  ...(End)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537

Crossrefs

Cf. A000120, A001222, A008687, A011782, A023416, A048881, A063787, A070939, A163511, A290252, A366275.

Programs

Maple

a := proc (n) if n < 2 then n elif `mod`(n, 2) = 0 then 1+a((1/2)*n) else a((1/2)*n-1/2) end if end proc: seq(a(n), n = 0 .. 150);

Mathematica

a[n_] := a[n] = Which[n < 2, n, EvenQ[n], 1+a[n/2], True, a[(n-1)/2]];
Table[a[n], {n, 0, 105}] (* Jean-François Alcover, Aug 06 2024 *)

PARI

A290251(n) = ((n>0)+#binary(n)-hammingweight(n)); \\ Antti Karttunen, Oct 06 2023

Formula

a(1) = 1; a(2n) = 1 + a(n); a(2n+1) = a(n).

a(n) = 1 + number of 0's in the binary form of n = 1 + A023416(n) for n>0.

a(n) = A008687(n+1) for n>0.

a(n) = 1 + A070939(n) - A000120(n) = A070939(n) - A048881(n+1). - Omar E. Pol, Jul 24 2017

a(n) = A001222(A163511(n)) = A001222(A366275(n)). - Antti Karttunen, Oct 06 2023

Extensions

Data section extended up to n=105 by Antti Karttunen, Oct 06 2023