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A170949 "Conway's Converger": a reordering of the integers (see Comments for definition). 6

%I

%S 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,

%T 33,31,29,27,26,28,30,32,34,36,48,46,44,42,40,38,37,39,41,43,45,47,49,

%U 63,61,59,57,55,53,51,50,52,54,56,58,60,62,64,80,78,76,74,72

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

%C 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.

%C 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]

%C From _Reinhard Zumkeller_, Mar 08 2010: (Start)

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

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

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

%D J. H. Conway, Personal communication, Feb 19 2010

%H R. Zumkeller, <a href="/A170949/b170949.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a> [From _Reinhard Zumkeller_, Mar 08 2010]

%e 1

%e 3 2 4

%e 8 6 5 7 9

%e 15 13 11 10 12 14 16

%e 24 22 20 18 17 19 21 23 25

%e 35 33 31 29 27 26 28 30 32 34 36

%e 48 46 44 42 40 38 37 39 41 43 45 47 49

%e 63 61 59 57 55 53 51 50 52 54 56 58 60 62 64

%e 80 78 76 74 72 70 68 66 65 67 69 71 73 75 77 79 81

%t row[n_] := Join[ro = Range[n^2-1, (n-1)^2+1, -2], Reverse[ro]-1, {n^2}];

%t Array[row, 9] // Flatten (* _Jean-Fran├žois Alcover_, Aug 02 2018 *)

%o (Haskell)

%o a170949 n k = a170949_tabf !! (n-1) !! (k-1)

%o a170949_row n = a170949_tabf !! (n-1)

%o a170949_tabf = [1] : (map fst $ iterate f ([3,2,4], 3)) where

%o f (xs@(x:_), i) = ([x + i + 2] ++ (map (+ i) xs) ++ [x + i + 3], i + 2)

%o a170949_list = concat a170949_tabf

%o -- _Reinhard Zumkeller_, Jan 31 2014

%Y Cf. A170950, A009858.

%Y Cf. A000290 (right diagonal), A132411 (left diagonal). - _Michel Marcus_, Aug 02 2018

%K nonn,tabf,look

%O 1,2

%A _N. J. A. Sloane_, Feb 21 2010

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Last modified January 22 18:06 EST 2019. Contains 319365 sequences. (Running on oeis4.)