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%I #43 Oct 06 2023 14:14:22
%S 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,
%T 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,
%U 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
%N a(n) is the number of parts in the integer partition having viabin number n.
%C 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.
%C From _Omar E. Pol_, Jul 24 2017: (Start)
%C 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).
%C Conjectures:
%C 1) It appears that if k > 1 and k is a power of 2 then column k lists the positive integers (A000027).
%C 2) It appears that column k lists the nonnegative integers (A001477) starting from the first elements of the column k.
%C 3) It appears that if n > 0 then row n lists the first 2^(n-1) elements of A063787 in reverse order. (End)
%H Antti Karttunen, <a href="/A290251/b290251.txt">Table of n, a(n) for n = 0..65537</a>
%F a(1) = 1; a(2n) = 1 + a(n); a(2n+1) = a(n).
%F a(n) = 1 + number of 0's in the binary form of n = 1 + A023416(n) for n>0.
%F a(n) = A008687(n+1) for n>0.
%F a(n) = 1 + A070939(n) - A000120(n) = A070939(n) - A048881(n+1). - _Omar E. Pol_, Jul 24 2017
%F a(n) = A001222(A163511(n)) = A001222(A366275(n)). - _Antti Karttunen_, Oct 06 2023
%e 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.
%e From _Omar E. Pol_, Jul 24 2017: (Start)
%e Written as an irregular triangle the sequence begins:
%e 0;
%e 1;
%e 2,1;
%e 3,2,2,1;
%e 4,3,3,2,3,2,2,1;
%e 5,4,4,3,4,3,3,2,4,3,3,2,3,2,2,1;
%e 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;
%e ...(End)
%p 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);
%o (PARI) A290251(n) = ((n>0)+#binary(n)-hammingweight(n)); \\ _Antti Karttunen_, Oct 06 2023
%Y Cf. A000120, A001222, A008687, A011782, A023416, A048881, A063787, A070939, A163511, A290252, A366275.
%K nonn,tabf
%O 0,3
%A _Emeric Deutsch_, Jul 24 2017
%E Data section extended up to n=105 by _Antti Karttunen_, Oct 06 2023
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