A254788 Lexicographically earliest permutation of the natural numbers, such that both numbers a(n)+a(n+1) and abs(a(n)-a(n+1)) occur all in all not more than once.

1, 2, 4, 8, 3, 10, 18, 9, 23, 6, 16, 31, 5, 25, 41, 7, 26, 44, 13, 36, 15, 39, 14, 49, 11, 48, 87, 12, 52, 93, 17, 60, 102, 19, 63, 108, 20, 66, 116, 21, 73, 128, 22, 78, 136, 24, 85, 147, 27, 92, 159, 28, 96, 165, 29, 100, 172, 30, 103, 177, 33, 113, 34

Offset

1,2

Comments

We want the sequence S to be a permutation of the positive integers.

We want S to be the lexicographically earliest sequence with these properties:

If we take two adjacent integers of S, say p & q, then:

- no other pair of adjacent integers in S has the same absolute difference |p-q|,

- no other pair of adjacent integers in S has the same sum (p+q),

- no |p-q|=(p'+q') with p'and q' being two other adjacent integers in S.

So S is extended with the smallest integer n such that neither |a(n-1)-a(n)|nor [a(n-1)+a(n)] has occurred before as a sum or as a difference of two adjacent integers in S.

A254792(n) = abs(a(n)-a(n+1));

A254793(n) = a(n) + a(n+1).

Links

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

Éric Angelini, Absolute diff and sums not to be shared, SeqFan list, Jan 15 2015.

Index entries for sequences that are permutations of the natural numbers

Example

The sequence begins like this, together with their pairwise sums and absolute differences:
. A254793(n):  3 6 12 11 13  28  27 32  29 22  47  36 30  66  48 33  70 ...
.       a(n): 1 2 4  8  3  10  18  9  23  6  16  31  5  25  41  7  26 44 ...
. A254792(n):  1 2  4  5  7   8   9 14  17 10  15  26 20  16  34 19  18 ...

Prog

(Haskell)
import Data.List (delete)
a254788 n = a254788_list !! (n-1)
a254788_list = 1 : f [2..] 1 [] where
   f xs y zs = g xs where
     g (w:ws) | s `elem` zs || d `elem` zs = g ws
              | otherwise = w : f (delete w xs) w (d : s : zs)
              where s = y + w; d = abs (y - w)

Crossrefs

Cf. A254790 (inverse), A254792, A254793.

Sequence in context: A319268 A277272 A109588 * A347669 A352812 A052331

Adjacent sequences: A254785 A254786 A254787 * A254789 A254790 A254791

Keyword

nonn

Author

Eric Angelini and Reinhard Zumkeller, Feb 07 2015

Status

approved