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A170949
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"Conway's Converger": a reordering of the integers (see Comments for definition).
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6
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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J. H. Conway, Personal communication, Feb 19 2010
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LINKS
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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]
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,tabf,look
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AUTHOR
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N. J. A. Sloane, Feb 21 2010
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STATUS
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approved
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