|
Search: id:A114994
|
|
|
| A114994 |
|
Numbers whose binary representation has monotonically decreasing sizes of groups of zeros (including zero-length groups between adjacent ones). |
|
+0
2
|
|
| 0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 21, 23, 31, 32, 33, 34, 35, 36, 37, 39, 42, 43, 47, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 79, 85, 87, 95, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139, 143, 146, 147, 149, 151, 159, 170, 171, 175
(list; graph; listen)
|
|
|
OFFSET
|
0,3
|
|
|
COMMENT
|
Numbers whose binary representation avoids the sequences 110, 10100, 1001000, etc. Represents partitions. Start with empty partition, and process each bit from left to right: if a zero, increase the size of the smallest part; if one, add a new size 1 part. This generates the partitions in Mathematica order. Can be regarded as a table with row lengths A000041(n); values 2^n <= a(m) < 2^(n+1) are in row n, representing the partitions of n. (Interpreting arbitrary binary numbers in this way generates compositions (aka ordered partitions); these are the compositions where the part sizes are in decreasing order of size.)
|
|
FORMULA
|
For n>=0, 2n+1 is in the sequence iff n is in the sequence. For n>0, 2n is in the sequence iff both n is the sequence and, for some k>=0, n is congruent to 2^k mod 4^(k+1).
|
|
EXAMPLE
|
21 is included, binary 10101 has group sizes 1,1,0; 22 is not, binary 10110 has group sizes 1,0,1, which includes an increase.
Applying bits of 21 in order gives sequence of partitions: [], [1], [2], [2,1], [2^2], [2^2,1], so 21 represents the partition [2^2,1].
|
|
CROSSREFS
|
Cf. A004743, A080577, A000041.
Sequence in context: A023748 A107686 A004743 this_sequence A137706 A039224 A039264
Adjacent sequences: A114991 A114992 A114993 this_sequence A114995 A114996 A114997
|
|
KEYWORD
|
easy,nonn,tabf
|
|
AUTHOR
|
Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 22 2006
|
|
|
Search completed in 0.002 seconds
|
