

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
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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 2n1 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]
Contribution from Reinhard Zumkeller, Mar 08 2010: (Start)
a(n + 2*A003059(n)) = a(n) + 2*A003059(n)  1;
a(A002522(n1)) = A132411(n); a(A002061(n)) = A002522(n1). (End)
The sum of the rows is n^3+(n+1)^3 [A005898] (1,9,35,91,189,...,) [From 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 [From Reinhard Zumkeller, Mar 08 2010]
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


PROG

(Haskell)
a170949 n k = a170949_tabf !! (n1) !! (k1)
a170949_row n = a170949_tabf !! (n1)
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.
Sequence in context: A254051 A082228 A114650 * A276953 A276943 A228784
Adjacent sequences: A170946 A170947 A170948 * A170950 A170951 A170952


KEYWORD

nonn,tabf,look


AUTHOR

N. J. A. Sloane, Feb 21 2010


STATUS

approved



