A234586 Odd-indexed terms are absolute values of differences.

1, 2, 1, 4, 3, 6, 1, 8, 3, 10, 5, 12, 7, 14, 5, 16, 7, 18, 5, 20, 7, 22, 5, 24, 7, 26, 9, 28, 11, 30, 9, 32, 11, 34, 13, 36, 15, 38, 13, 40, 15, 42, 17, 44, 19, 46, 17, 48, 19, 50, 17, 52, 19, 54, 17, 56, 19, 58, 21, 60, 23, 62, 21, 64, 23, 66, 21, 68, 23, 70, 21, 72, 23, 74, 25, 76, 27, 78, 25, 80, 27, 82, 25

Offset

1,2

Comments

The precise definition is: Set a(2n)=2n for all n, set a(1)=1, and for n >= 1 choose a(2n+1) so that the subsequence {a(2i+1), i>=0} is the same as the sequence of differences {|a(j+1)-a(j)|, j>=0}.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Eric Angelini, The first differences of S ...

Eric Angelini, The first differences of S ... [Cached copy, with permission]

Formula

There is a surprising connection with the Thue-Morse sequence A010060. If the k-th run of equal terms in A010060 (k>=0) has length L (L=1 or 2, see A026465), replace it by 2L copies of the pair 4k+1, 4k+3. This produces the odd-indexed terms of the sequence (ignoring the initial 1): 0 1 1 0 1 0 0 1 ... becomes 1 3 1 3 5 7 5 7 5 7 5 7 9 11 9 11 13 ... - N. J. A. Sloane, Dec 31 2013

Example

We start by alternating even numbers and "holes" like this:
S = . 2 . 4 . 6 . 8 . 10 . 12 . 14 . 16 . 18 . 20 . 22 .....
We fill the first hole with '1' and the second and third holes with x, y:
S = 1 2 x 4 y 6 . 8 . 10 . 12 . 14 . 16 . 18 . 20 . 22 .....
The absolute values of differences are 1, |x-2|, |4-x|, ... which must equal 1, x, y, ..., which forces x=1, y=3. And so on.

Maple

with(LinearAlgebra): M:=1000; S:=Array(1..2*M); S[1]:=1; S[3]:=1;
for i from 1 to M do S[2*i]:=2*i; od:
for i from 2 to M-1 do S[2*i+1]:=abs(S[i+2]-S[i+1]); od:
[seq(S[i], i=1..2*M)];

Mathematica

a[1] = a[3] = 1; a[n_?EvenQ] := n; a[n_] := a[n] = Abs[a[(n-1)/2+2]-a[(n-1)/2+1]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 13 2015 *)

Prog

(Haskell)
import Data.List (transpose)
a234586 n = a234586_list !! (n-1)
a234586_list = concat (transpose [a234587_list, [2, 4 ..]])
a234587_list = 1 : 1 : (drop 2 $
               map abs $ zipWith (-) a234586_list $ tail a234586_list)
-- Reinhard Zumkeller, Jul 15 2014

Crossrefs

Cf. A010060, A026465, A234587 (the odd-indexed terms).

Sequence in context: A075348 A326062 A055631 * A338657 A347082 A186421

Adjacent sequences: A234583 A234584 A234585 * A234587 A234588 A234589

Keyword

nonn,nice

Author

Eric Angelini, Dec 31 2013

Extensions

Entered by N. J. A. Sloane on Eric Angelini's behalf and submitted for the 2014 JMM competition with his permission.

Status

approved