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

%I A290251 #47 Aug 06 2024 09:57:08
%S A290251 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 A290251 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 A290251 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 A290251 a(n) is the number of parts in the integer partition having viabin number n.
%C A290251 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 A290251 From _Omar E. Pol_, Jul 24 2017: (Start)
%C A290251 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 A290251 Conjectures:
%C A290251 1) It appears that if k > 1 and k is a power of 2 then column k lists the positive integers (A000027).
%C A290251 2) It appears that column k lists the nonnegative integers (A001477) starting from the first elements of the column k.
%C A290251 3) It appears that if n > 0 then row n lists the first 2^(n-1) elements of A063787 in reverse order. (End)
%H A290251 Antti Karttunen, <a href="/A290251/b290251.txt">Table of n, a(n) for n = 0..65537</a>
%F A290251 a(1) = 1; a(2n) = 1 + a(n); a(2n+1) = a(n).
%F A290251 a(n) = 1 + number of 0's in the binary form of n = 1 + A023416(n) for n>0.
%F A290251 a(n) = A008687(n+1) for n>0.
%F A290251 a(n) = 1 + A070939(n) - A000120(n) = A070939(n) - A048881(n+1). - _Omar E. Pol_, Jul 24 2017
%F A290251 a(n) = A001222(A163511(n)) = A001222(A366275(n)). - _Antti Karttunen_, Oct 06 2023
%e A290251 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 A290251 From _Omar E. Pol_, Jul 24 2017: (Start)
%e A290251 Written as an irregular triangle the sequence begins:
%e A290251   0;
%e A290251   1;
%e A290251   2,1;
%e A290251   3,2,2,1;
%e A290251   4,3,3,2,3,2,2,1;
%e A290251   5,4,4,3,4,3,3,2,4,3,3,2,3,2,2,1;
%e A290251   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 A290251   ...(End)
%p A290251 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);
%t A290251 a[n_] := a[n] = Which[n < 2, n, EvenQ[n], 1+a[n/2], True, a[(n-1)/2]];
%t A290251 Table[a[n], {n, 0, 105}] (* _Jean-François Alcover_, Aug 06 2024 *)
%o A290251 (PARI) A290251(n) = ((n>0)+#binary(n)-hammingweight(n)); \\ _Antti Karttunen_, Oct 06 2023
%Y A290251 Cf. A000120, A001222, A008687, A011782, A023416, A048881, A063787, A070939, A163511, A290252, A366275.
%K A290251 nonn,tabf
%O A290251 0,3
%A A290251 _Emeric Deutsch_, Jul 24 2017
%E A290251 Data section extended up to n=105 by _Antti Karttunen_, Oct 06 2023