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Search: id:A126441
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| A126441 |
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Tabular arrangement of the natural numbers: the row on which any non-zero term a(n) appears in is A053645(a(n))=A053645(n+1), and the column is A161511(a(n)). Table is presented by columns with 2^{k-1} items in column k, unused positions are filled with 0's. |
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10
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| 1, 2, 3, 4, 5, 0, 7, 8, 9, 6, 11, 0, 0, 0, 15, 16, 17, 10, 19, 0, 13, 0, 23, 0, 0, 0, 0, 0, 0, 0, 31, 32, 33, 18, 35, 12, 21, 14, 39, 0, 0, 0, 27, 0, 0, 0, 47, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 63, 64, 65, 34, 67, 20, 37, 22, 71, 0, 25, 0, 43, 0, 29, 0, 79, 0, 0, 0, 0, 0, 0, 0, 55, 0, 0
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Note: 1 might be a more natural starting offset for this sequence, although the identities concerning A053645 and A161511 would have to be changed. - Antti Karttunen, Oct 12 2009.
This can be regarded as an arrangement of the partitions, indexed by position in A125106. The partitions in a given row all have the same remaining partition when the largest part is removed; specifically, the partition indexed by the row number in A125106 (with row 0 having the empty partition remaining).
The first value on row n is A004760(n+1). The second value on each row is A004760(n+1) plus A062383(n); subsequent values increase by ever enlarging powers of two. Or equivalently, each subsequent value on the row after the first non-zero value is given by A004754(previous value on the same row).
A055941(r) tells how many terms the row r (>= 0) has been shifted rightward from its "natural position", i.e. with how many zeros that row has been prepended.
The number of (non-zero) entries in column k is A000041(k).
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LINKS
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A. Karttunen, Table of n, a(n) for n = 0..65534 (first 16 columns)
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EXAMPLE
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The largest power of 2 <= 6 is 4, 6 - 4 = 2, so 6 is in row 2. By A125106, 6 corresponds to the partition [2^2], total 4, so 6 goes in column 4. Thus T(2,4) = 6.
The table begins:
1.2.4..8.16.32.64.128.256.512.1024
..3.5..9.17.33.65.129.257.513.1025
.......6.10.18.34..66.130.258..514
....7.11.19.35.67.131.259.515.1027
............12.20..36..68.132..260
.........13.21.37..69.133.261..517
............14.22..38..70.134..262
......15.23.39.71.135.263.519.1031
...................24..40..72..136
...............25..41..73.137..265
...................26..42..74..138
............27.43..75.139.267..523
.......................28..44...76
...............29..45..77.141..269
...................30..46..78..142
.........31.47.79.143.271.527.1039
...........................48...80
.......................49..81..145
...........................50...82
...................51..83.147..275
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PROGRAM
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(GNU/MIT Scheme:)
(define (A126441 n) (A126441onebased (1+ n)))
(definec (A126441onebased n) (cond ((< n 2) n) (else (let ((prev (A126441onebased (- n (/ (A053644 n) 2))))) (if (or (= (A053644 n) (* 2 (A053644 (A053645 n)))) (zero? prev)) (let ((starter (A004760 (1+ (A053645 n))))) (if (> (A161511 starter) (1+ (A000523 n))) 0 starter)) (A004754 prev))))))
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CROSSREFS
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Cf. A125106, A053645, A000041, A004760, A062383, A000079 (column lengths).
A053645(a(A166274(n))) = A053645(1+A166274(n)) for all n>=1.
Positions of zeros: A166275, this sequence without zeros: A161924. A161920(n) gives the position of the first non-zero term on the row n-1.
Sequence in context: A095874 A063972 A063973 this_sequence A004181 A080744 A030548
Adjacent sequences: A126438 A126439 A126440 this_sequence A126442 A126443 A126444
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KEYWORD
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nonn,tabf
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AUTHOR
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Alford Arnold (Alford1940(AT)aol.com), Jan 19 2007
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EXTENSIONS
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Edited by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Jan 23 2007. Further edited and Scheme-code added by Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 12 2009
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