A170949 "Conway's Converger": a reordering of the integers (see Comments for definition).

1, 3, 2, 4, 8, 6, 5, 7, 9, 15, 13, 11, 10, 12, 14, 16, 24, 22, 20, 18, 17, 19, 21, 23, 25, 35, 33, 31, 29, 27, 26, 28, 30, 32, 34, 36, 48, 46, 44, 42, 40, 38, 37, 39, 41, 43, 45, 47, 49, 63, 61, 59, 57, 55, 53, 51, 50, 52, 54, 56, 58, 60, 62, 64, 80, 78, 76, 74, 72

Offset: 1

Author

N. J. A. Sloane, Feb 21 2010

Keywords

• nonn
• tabf
• look

Comments

The integers are written in blocks of lengths 1, 3, 5, 7, 9, ... . The first number in the block is moved to the center of the block, and then the numbers are written alternately to the left and the right. The block of length 2n-1 ends with n^2, which is not moved.

Let S = Sum_{i >= 1} s(i) be a not necessarily converging series and let T = Sum_{i >= 1} s(a(i)). Then if S converges so does T. On the other hand there are examples where T converges but S does not (for example S = 1 + 1 + 0 - 1 + 1/2 + 1/2 + 0 - 1/2 - 1/2 + 1/3 (3 times) + 0 - 1/3 (3 times) + 1/5 (5 times) + 0 - 1/5 (5 times) + ...). [Conway]

From Reinhard Zumkeller, Mar 08 2010: (Start)

a(n + 2*A003059(n)) = a(n) + 2*A003059(n) - 1;

a(A002522(n-1)) = A132411(n); a(A002061(n)) = A002522(n-1). (End)

The sum of the rows is n^3+(n+1)^3 [A005898] (1,9,35,91,189,...). - Vincenzo Librandi, Feb 22 2010

Examples

			                           1
                        3  2  4
                     8  6  5  7  9
                 15 13 11 10 12 14 16
              24 22 20 18 17 19 21 23 25
           35 33 31 29 27 26 28 30 32 34 36
        48 46 44 42 40 38 37 39 41 43 45 47 49
     63 61 59 57 55 53 51 50 52 54 56 58 60 62 64
  80 78 76 74 72 70 68 66 65 67 69 71 73 75 77 79 81
		

References

• J. H. Conway, Personal communication, Feb 19 2010

Links

• R. Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for sequences that are permutations of the natural numbers [From _Reinhard Zumkeller_, Mar 08 2010]

Crossrefs

Cf. A170950, A009858.

Cf. A000290 (right diagonal), A132411 (left diagonal). - Michel Marcus, Aug 02 2018

Programs

Haskell

a170949 n k = a170949_tabf !! (n-1) !! (k-1)
a170949_row n = a170949_tabf !! (n-1)
a170949_tabf = [1] : (map fst $ iterate f ([3,2,4], 3)) where
  f (xs@(x:_), i) = ([x + i + 2] ++ (map (+ i) xs) ++ [x + i + 3], i + 2)
a170949_list = concat a170949_tabf
-- Reinhard Zumkeller, Jan 31 2014

Mathematica

row[n_] := Join[ro = Range[n^2-1, (n-1)^2+1, -2], Reverse[ro]-1, {n^2}];
Array[row, 9] // Flatten (* Jean-François Alcover, Aug 02 2018 *)