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 A170949 "Conway's Converger": a reordering of the integers (see Comments for definition). 6
 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 (list; graph; refs; listen; history; text; internal format)
 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 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 =  : (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

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Last modified June 16 21:55 EDT 2021. Contains 345080 sequences. (Running on oeis4.)