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

1, 3, 6, 10, 14, 18, 23, 29, 35, 41, 47, 53, 60, 68, 76, 84, 92, 100, 108, 116, 125, 135, 145, 155, 165, 175, 185, 195, 205, 215, 226, 238, 250, 262, 274, 286, 298, 310, 322, 334, 346, 358, 371, 385, 399, 413, 427, 441, 455, 469, 483, 497, 511, 525, 539, 553

Offset

1,2

Comments

The natural numbers are filled into square blocks of edge length 2, 4, 6, 8, ...

by taking A016742(n+1) = 4, 16, 36, ... at a time:

.......1..2......

.......3..4......

....5..6..7..8...

....9.10.11.12...

...13.14.15.16...

...17.18.19.20...

21.22.23.24.25.26

27.28.29.30.31.32

33.34.35.36.37.38

39.40.41.42.43.44

Reading down the column just left from the center yields a(n).

The length of the rows is given by A001670.

The number of elements in each square block, 4, 16, 36, etc., are the first differences of A166464:

A016742(n) = A166464(n)-A166464(n-1).

Reading the blocks from right to left, row by row, we obtain a permutation of the integers, which starts similar to A166133.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Mathematica

r[1] = Range[4]; r[n_] := r[n] = Range[r[n-1][[-1]]+1, r[n-1][[-1]]+(2n)^2 ];
s[n_] := Partition[r[n], Sqrt[Length[r[n]]]][[All, n]];
A167381 = Table[s[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Mar 26 2017 *)
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 *)

Crossrefs

Cf. A113127, A167991 (first differences).

Sequence in context: A113127 A145913 A130246 * A269745 A310065 A310066

Adjacent sequences: A167378 A167379 A167380 * A167382 A167383 A167384

Keyword

nonn,easy

Author

Paul Curtz, Nov 02 2009

Extensions

Edited by R. J. Mathar, Aug 29 2010

More terms from Jean-François Alcover, Mar 26 2017

Status

approved