A290254 The viabin numbers of the self-conjugate integer partitions.

0, 1, 5, 6, 19, 21, 26, 28, 71, 75, 85, 89, 102, 106, 116, 120, 271, 279, 299, 307, 333, 341, 361, 369, 398, 406, 426, 434, 460, 468, 488, 496, 1055, 1071, 1111, 1127, 1179, 1195, 1235, 1251, 1309, 1325, 1365, 1381, 1433, 1449, 1489, 1505, 1566, 1582, 1622

Offset

1,3

Comments

For the definition of viabin number see comment in A290253.

The binary representation of a(n) is obtained by concatenating the binary representation of (n-1) and the reversed bit-flipped binary representation of (n-1) and then dropping the last bit. This suggests that it would have been more natural to index from 0. - Peter J. Taylor, Sep 24 2021

Links

Alois P. Heinz, Table of n, a(n) for n = 1..16384

Formula

{ 0 } union fixed points of A059894. - Alois P. Heinz, Aug 24 2017

a(n) = a(n-1) + 2*4^(f(n-1) - 1) + 3*2^(f(n-1) - 1) - 1 if n = 2^k + 1, k > 0, otherwise a(n-1) + (2^(A007814(n-1) + 2) - 3)*2^f(A025480(n-2)) with a(1) = 0, a(2) = 1 where f(n) = A000523(n) for n > 0 with f(0) = 0. - Mikhail Kurkov, Sep 24 2021

Example

19 is in the sequence. Indeed, binary (19) = 10011 and so the southeast border of the Ferrers board of the corresponding integer partition is ENNEEN, where E = (1,0) and N = (0,1). This leads to the self-conjugate integer partition [3,1,1].

Maple

a := proc (n) local i, m, r: m, r := n, 0: for i from 0 while 1 < m do r := 2*r+1-irem(m, 2, 'm') end do: r+2^i end proc: SC := {0}: for n to 3000 do if a(n) = n then SC := `union`(SC, {n}) else  end if end do: SC; # first part of the program taken from A059894.

Mathematica

nmax = 3000; (* nmax=3000 gives 64 terms *)
a[n_] := Module[{i, m = n, r = 0}, For[i = 0, 1 < m, i++, r = 2*r + 1 - Mod[m, 2]; m = Quotient[m, 2]]; r + 2^i];
SC = {0};
For[n = 1, n <= nmax, n++, If[a[n] == n, SC = Union[SC, {n}]]];
SC (* Jean-François Alcover, Dec 16 2020, after Maple *)

Prog

(Python) a = lambda n: int(bin(n-1)[2:] + ''.join(str(1 ^ int(ch)) for ch in bin(n-1)[-1:2:-1]), 2) # Peter J. Taylor, Sep 24 2021
(PARI) a(n) = my(v=binary(max(1, n-1))[^1]); n<<#v + bitneg(fromdigits(Vecrev(v), 2)); \\ Kevin Ryde, Nov 30 2021

Crossrefs

Cf. A059894, A290253.

Sequence in context: A257331 A240400 A031448 * A163772 A056509 A129722

Adjacent sequences: A290251 A290252 A290253 * A290255 A290256 A290257

Keyword

nonn

Author

Emeric Deutsch, Aug 23 2017

Status

approved