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A194602 Integer partitions interpreted as binary numbers. 13
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.

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 follow from the fact that the first p(n) terms also correspond to the list of nondecreasing compositions of n, and those of (n+1) with at least one part of size 1, in lexicographic order.

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

Li-yao Xia, Identities for A194602

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 nondecreasing 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. A210941 (triangle of the non-one addends).

Cf. A194548 (numbers of non-one addends, i.e., row lengths of A210941).

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

EXTENSIONS

Comments edited by Li-yao Xia, May 13 2014

STATUS

approved

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Last modified September 2 15:46 EDT 2014. Contains 246361 sequences.