This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A126441 #16 Dec 13 2023 18:29:05
%S A126441 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,
%T A126441 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,
%U A126441 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
%N A126441 Tabular arrangement of the natural numbers: the row on which any nonzero 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.
%C A126441 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.
%C A126441 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).
%C A126441 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 nonzero value is given by A004754(previous value on the same row).
%C A126441 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.
%C A126441 The number of (nonzero) entries in column k is A000041(k).
%H A126441 A. Karttunen, <a href="/A126441/b126441.txt">Table of n, a(n) for n = 0..65534 (first 16 columns)</a>
%e A126441 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.
%e A126441 The table begins:
%e A126441 1.2.4..8.16.32.64.128.256.512.1024
%e A126441 ..3.5..9.17.33.65.129.257.513.1025
%e A126441 .......6.10.18.34..66.130.258..514
%e A126441 ....7.11.19.35.67.131.259.515.1027
%e A126441 ............12.20..36..68.132..260
%e A126441 .........13.21.37..69.133.261..517
%e A126441 ............14.22..38..70.134..262
%e A126441 ......15.23.39.71.135.263.519.1031
%e A126441 ...................24..40..72..136
%e A126441 ...............25..41..73.137..265
%e A126441 ...................26..42..74..138
%e A126441 ............27.43..75.139.267..523
%e A126441 .......................28..44...76
%e A126441 ...............29..45..77.141..269
%e A126441 ...................30..46..78..142
%e A126441 .........31.47.79.143.271.527.1039
%e A126441 ...........................48...80
%e A126441 .......................49..81..145
%e A126441 ...........................50...82
%e A126441 ...................51..83.147..275
%t A126441 columns = 7; row[n_] := n-2^Floor[Log2[n]]; col[0] = 0; col[n_] := If[EvenQ[n], col[n/2] + DigitCount[n/2, 2, 1], col[(n-1)/2]+1]; Clear[T]; T[_, _] = 0; Do[T[row[k], col[k]] = k, {k, 1, 2^columns}]; Table[T[n-1, k], {k, 1, columns}, {n, 1, 2^(k-1)}] // Flatten (* _Jean-François Alcover_, Sep 09 2017 *)
%o A126441 (MIT/GNU Scheme)
%o A126441 (define (A126441 n) (A126441onebased (1+ n)))
%o A126441 (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))))))
%Y A126441 Cf. A125106, A053645, A000041, A004760, A062383, A000079 (column lengths).
%Y A126441 A053645(a(A166274(n))) = A053645(1+A166274(n)) for all n>=1.
%Y A126441 Positions of zeros: A166275, this sequence without zeros: A161924. A161920(n) gives the position of the first nonzero term on the row n-1.
%K A126441 nonn,tabf
%O A126441 0,2
%A A126441 _Alford Arnold_, Jan 19 2007
%E A126441 Edited by _Franklin T. Adams-Watters_, Jan 23 2007
%E A126441 Further edited and Scheme-code added by _Antti Karttunen_, Oct 12 2009