A352750 a(n) = binary complement of b(n-1) AND b(n); a(1) = 1, where b(n) = A109812(n).

1, 0, 1, 0, 4, 2, 0, 5, 9, 0, 4, 1, 2, 0, 17, 24, 0, 6, 10, 0, 16, 8, 4, 2, 8, 16, 2, 0, 48, 41, 0, 20, 36, 0, 24, 36, 0, 18, 32, 8, 4, 32, 24, 0, 32, 16, 3, 8, 0, 39, 98, 0, 28, 96, 0, 28, 80, 0, 41, 81, 16, 8, 4, 2, 1, 4, 8, 1, 0, 48, 68, 0, 48, 64, 3, 32, 64

Offset

1,5

Comments

b(n) = A109812(n) is the least k that does not appear in b(1..n-1) that avoids the ON bits in b(n-1).

a(n) is the decimal value of available bits "unused" in k.

If b(n) = 2^m - 1, then a(n) = 0 iff b(n) = 2^m.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Log-log scatterplot of a(n) for n = 1..2^16, substituting 1/2 for 0 so as to plot in blue, records labelled in red, fixed points shown in amber.

Michael De Vlieger, Bitmap of a(n) for n = 1..2^10, where each term is shown vertically, with least significant bit at bottom, 12X vertical exaggeration. Black corresponds to 1 and white to 0.

Michael De Vlieger, Bitmap of a(n) for n = 1..2^14, where each term is shown horizontally, with least significant bit at right, 256X horizontal exaggeration. Black corresponds to 1 and white to 0.

Rémy Sigrist, PARI program

Formula

a(n) = 2^(1 + max(floor(log_2 i), floor(log_2 j))) - (i + j) - 1, with i = A109812(n-1) and j = A109812(n). - Michael De Vlieger, Apr 05 2022

Example

Relation between a(n) and b(n) = A109812(n). The binary expansion of b(n) = b(n)_2 = A352575(n), showing zeros as "." to accentuate 1 bits:
n    b(n)_2  b(n) a(n)
----------------------
1         1    1    1
2        1.    2    0
3       1..    4    1
4        11    3    0
5      1...    8    4
6       1.1    5    2
7      1.1.   10    0
8     1....   16    5
9       11.    6    9
10     1..1    9    0
11    1..1.   18    4
12     11..   12    1
13    1...1   17    2
14     111.   14    0
15   1.....   32   17
16      111    7   24
...

Mathematica

c[_] = 0; a[1] = c[1] = 1; u = 2; {1}~Join~Reap[Do[k = u; While[Nand[c[k] == 0, BitAnd[a[i - 1], k] == 0], k++]; Set[{a[i], c[k]}, {k, i}]; Sow[FromDigits[IntegerDigits[BitOr[a[i - 1], a[i]], 2] /. {0 -> 1, 1 -> 0}, 2]]; If[k == u, While[c[u] > 0, u++]], {i, 2, 77}]][[-1, -1]]

Prog

(PARI) See Links section.

Crossrefs

Cf. A109812, A352575.

Sequence in context: A135730 A188595 A144102 * A195388 A255324 A328819

Adjacent sequences: A352747 A352748 A352749 * A352751 A352752 A352753

Keyword

nonn,look

Author

Michael De Vlieger, Apr 01 2022

Status

approved