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A167381 The numbers read down the left-center column of an arrangement of the natural numbers in square blocks.

%I A167381 #19 Oct 18 2018 16:12:26
%S A167381 1,3,6,10,14,18,23,29,35,41,47,53,60,68,76,84,92,100,108,116,125,135,
%T A167381 145,155,165,175,185,195,205,215,226,238,250,262,274,286,298,310,322,
%U A167381 334,346,358,371,385,399,413,427,441,455,469,483,497,511,525,539,553
%N A167381 The numbers read down the left-center column of an arrangement of the natural numbers in square blocks.
%C A167381 The natural numbers are filled into square blocks of edge length 2, 4, 6, 8, ...
%C A167381 by taking A016742(n+1) = 4, 16, 36, ... at a time:
%C A167381 .......1..2......
%C A167381 .......3..4......
%C A167381 ....5..6..7..8...
%C A167381 ....9.10.11.12...
%C A167381 ...13.14.15.16...
%C A167381 ...17.18.19.20...
%C A167381 21.22.23.24.25.26
%C A167381 27.28.29.30.31.32
%C A167381 33.34.35.36.37.38
%C A167381 39.40.41.42.43.44
%C A167381 Reading down the column just left from the center yields a(n).
%C A167381 The length of the rows is given by A001670.
%C A167381 The number of elements in each square block, 4, 16, 36, etc., are the first differences of A166464:
%C A167381 A016742(n) = A166464(n)-A166464(n-1).
%C A167381 Reading the blocks from right to left, row by row, we obtain a permutation of the integers, which starts similar to A166133.
%H A167381 Harvey P. Dale, <a href="/A167381/b167381.txt">Table of n, a(n) for n = 1..10000</a>
%t A167381 r[1] = Range[4]; r[n_] := r[n] = Range[r[n-1][[-1]]+1, r[n-1][[-1]]+(2n)^2 ];
%t A167381 s[n_] := Partition[r[n], Sqrt[Length[r[n]]]][[All, n]];
%t A167381 A167381 = Table[s[n], {n, 1, 7}] // Flatten (* _Jean-François Alcover_, Mar 26 2017 *)
%t A167381 Module[{nn=7,c},c=TakeList[Range[(2/3)*nn(nn+1)(2*nn+1)],(2*Range[ nn])^2]; Table[Take[c[[n]],{n,-1,2*n}],{n,nn}]]//Flatten (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Oct 18 2018 *)
%Y A167381 Cf. A113127, A167991 (first differences).
%K A167381 nonn,easy
%O A167381 1,2
%A A167381 _Paul Curtz_, Nov 02 2009
%E A167381 Edited by _R. J. Mathar_, Aug 29 2010
%E A167381 More terms from _Jean-François Alcover_, Mar 26 2017