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

%I A170949 #25 Aug 02 2018 04:15:48
%S A170949 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 A170949 33,31,29,27,26,28,30,32,34,36,48,46,44,42,40,38,37,39,41,43,45,47,49,
%U A170949 63,61,59,57,55,53,51,50,52,54,56,58,60,62,64,80,78,76,74,72
%N A170949 "Conway's Converger": a reordering of the integers (see Comments for definition).
%C A170949 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 A170949 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 A170949 From _Reinhard Zumkeller_, Mar 08 2010: (Start)
%C A170949 a(n + 2*A003059(n)) = a(n) + 2*A003059(n) - 1;
%C A170949 a(A002522(n-1)) = A132411(n); a(A002061(n)) = A002522(n-1). (End)
%C A170949 The sum of the rows is n^3+(n+1)^3 [A005898] (1,9,35,91,189,...). - _Vincenzo Librandi_, Feb 22 2010
%D A170949 J. H. Conway, Personal communication, Feb 19 2010
%H A170949 R. Zumkeller, <a href="/A170949/b170949.txt">Table of n, a(n) for n = 1..10000</a>
%H A170949 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a> [From _Reinhard Zumkeller_, Mar 08 2010]
%e A170949                            1
%e A170949                         3  2  4
%e A170949                      8  6  5  7  9
%e A170949                  15 13 11 10 12 14 16
%e A170949               24 22 20 18 17 19 21 23 25
%e A170949            35 33 31 29 27 26 28 30 32 34 36
%e A170949         48 46 44 42 40 38 37 39 41 43 45 47 49
%e A170949      63 61 59 57 55 53 51 50 52 54 56 58 60 62 64
%e A170949   80 78 76 74 72 70 68 66 65 67 69 71 73 75 77 79 81
%t A170949 row[n_] := Join[ro = Range[n^2-1, (n-1)^2+1, -2], Reverse[ro]-1, {n^2}];
%t A170949 Array[row, 9] // Flatten (* _Jean-François Alcover_, Aug 02 2018 *)
%o A170949 (Haskell)
%o A170949 a170949 n k = a170949_tabf !! (n-1) !! (k-1)
%o A170949 a170949_row n = a170949_tabf !! (n-1)
%o A170949 a170949_tabf = [1] : (map fst $ iterate f ([3,2,4], 3)) where
%o A170949   f (xs@(x:_), i) = ([x + i + 2] ++ (map (+ i) xs) ++ [x + i + 3], i + 2)
%o A170949 a170949_list = concat a170949_tabf
%o A170949 -- _Reinhard Zumkeller_, Jan 31 2014
%Y A170949 Cf. A170950, A009858.
%Y A170949 Cf. A000290 (right diagonal), A132411 (left diagonal). - _Michel Marcus_, Aug 02 2018
%K A170949 nonn,tabf,look
%O A170949 1,2
%A A170949 _N. J. A. Sloane_, Feb 21 2010