|
| |
|
|
A194602
|
|
Integer partitions interpreted as binary numbers.
|
|
10
|
|
|
|
0, 1, 3, 5, 7, 11, 15, 21, 23, 27, 31, 43, 47, 55, 63, 85, 87, 91, 95, 111, 119, 127, 171, 175, 183, 191, 219, 223, 239, 255, 341, 343, 347, 351, 367, 375, 383, 439, 447, 479, 495, 511, 683, 687, 695, 703, 731, 735, 751, 767, 879, 887, 895, 959, 991, 1023, 1365, 1367, 1371, 1375, 1391
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
The 2^(n-1) compositions of n correspond to binary numbers, and the partitions of n can be seen as compositions with addends ordered by size, so they also correspond to binary numbers.
The finite sequence for partitions of n (ordered by size) is the beginning of the sequence for partitions of n+1, which leads to an infinite sequence.
Contribution from Tilman Piesk, Apr 16 2012: (Start)
a( p(2n-1) ) = a( 1,3,7,15,30,56,101... ) = 1,5,21,85,341,1365,5461... = A002450(n) = binary numbers 1,101,10101,1010101... corresponding to partitions that have n addends 2 and no other non-one addends (from n=1).
a( p(n)-1 ) = a( 1,2,4,6,10,14,21... ) = 1,3,7,15,31,63,127... = A000225(n-1) = binary numbers 1,11,111,1111... corresponding to partitions with exactly one addend n and no other non-one addends (from n=2).
a( p(2n)-2 ) = a( 3,9,20,40,75,133... ) = 5,27,119,495,2015,8127... = A129868(n-1) = binary numbers 101,11011,1110111,111101111... corresponding to partitions with exactly two addends n and no other non-one addends (from n=2).
a( p(3n) -A004526(n) -3 ) = a( 7,26,72,171,379... ) = 21,219,1911,15855,128991... = binary numbers 10101,11011011,11101110111,11110111101111... corresponding to partitions with exactly three addends n and no other non-one addends (from n=2).
a is this sequence. p is the sequence of partition numbers A000041. These formulas are not proved but depend on observation for a number of entries that makes coincidence very unlikely. However, the last formula may deserve a certain amount of distrust. (End)
|
|
|
LINKS
|
Tilman Piesk, Table of n, a(n) for n = 0..8348
Tilman Piesk, Same table with binary strings and non-one addends
Tilman Piesk, Partitions of 10
|
|
|
EXAMPLE
|
From Joerg Arndt, Nov 17 2012: (Start)
With leading zeros included, the first A000041(n) terms correspond to the list of partitions of n as non-decreasing compositions in lexicographic order.
For example, the first A000041(10)=42 terms correspond to the partitions of 10 as follows (dots for zeros in the binary expansions):
[ n] binary(a(n)) a(n) partition
[ 0] .......... 0 [ 1 1 1 1 1 1 1 1 1 1 ]
[ 1] .........1 1 [ 1 1 1 1 1 1 1 1 2 ]
[ 2] ........11 3 [ 1 1 1 1 1 1 1 3 ]
[ 3] .......1.1 5 [ 1 1 1 1 1 1 2 2 ]
[ 4] .......111 7 [ 1 1 1 1 1 1 4 ]
[ 5] ......1.11 11 [ 1 1 1 1 1 2 3 ]
[ 6] ......1111 15 [ 1 1 1 1 1 5 ]
[ 7] .....1.1.1 21 [ 1 1 1 1 2 2 2 ]
[ 8] .....1.111 23 [ 1 1 1 1 2 4 ]
[ 9] .....11.11 27 [ 1 1 1 1 3 3 ]
[10] .....11111 31 [ 1 1 1 1 6 ]
[11] ....1.1.11 43 [ 1 1 1 2 2 3 ]
[12] ....1.1111 47 [ 1 1 1 2 5 ]
[13] ....11.111 55 [ 1 1 1 3 4 ]
[14] ....111111 63 [ 1 1 1 7 ]
[15] ...1.1.1.1 85 [ 1 1 2 2 2 2 ]
[16] ...1.1.111 87 [ 1 1 2 2 4 ]
[17] ...1.11.11 91 [ 1 1 2 3 3 ]
[18] ...1.11111 95 [ 1 1 2 6 ]
[19] ...11.1111 111 [ 1 1 3 5 ]
[20] ...111.111 119 [ 1 1 4 4 ]
[21] ...1111111 127 [ 1 1 8 ]
[22] ..1.1.1.11 171 [ 1 2 2 2 3 ]
[23] ..1.1.1111 175 [ 1 2 2 5 ]
[24] ..1.11.111 183 [ 1 2 3 4 ]
[25] ..1.111111 191 [ 1 2 7 ]
[26] ..11.11.11 219 [ 1 3 3 3 ]
[27] ..11.11111 223 [ 1 3 6 ]
[28] ..111.1111 239 [ 1 4 5 ]
[29] ..11111111 255 [ 1 9 ]
[30] .1.1.1.1.1 341 [ 2 2 2 2 2 ]
[31] .1.1.1.111 343 [ 2 2 2 4 ]
[32] .1.1.11.11 347 [ 2 2 3 3 ]
[33] .1.1.11111 351 [ 2 2 6 ]
[34] .1.11.1111 367 [ 2 3 5 ]
[35] .1.111.111 375 [ 2 4 4 ]
[36] .1.1111111 383 [ 2 8 ]
[37] .11.11.111 439 [ 3 3 4 ]
[38] .11.111111 447 [ 3 7 ]
[39] .111.11111 479 [ 4 6 ]
[40] .1111.1111 495 [ 5 5 ]
[41] .111111111 511 [ 10 ]
(End)
|
|
|
CROSSREFS
|
Cf. A000041 (partition numbers)
Cf. A002450, A000225, A129868 (subsequences)
Cf. A004526 (repeated non-negative integers)
Sequence in context: A219655 A007665 A208994 * A177139 A062488 A116582
Adjacent sequences: A194599 A194600 A194601 * A194603 A194604 A194605
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Tilman Piesk, Aug 30 2011
|
|
|
STATUS
|
approved
|
| |
|
|
