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

1,2

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

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]

Example

                           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

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 *)

Prog

(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

Crossrefs

Cf. A170950, A009858.

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

Sequence in context: A254051 A082228 A114650 * A276953 A276943 A303148

Adjacent sequences: A170946 A170947 A170948 * A170950 A170951 A170952

Keyword

nonn,tabf,look

Author

N. J. A. Sloane, Feb 21 2010

Status

approved