login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A229543 Number of undirected circular permutations i_0, i_1, ..., i_n of 0, 1, ..., n such that all the n+1 adjacent distances |i_0-i_1|, |i_1-i_2|, ..., |i_{n-1}-i_n|, |i_n-i_0| are perfect squares. 0
1, 0, 0, 1, 0, 1, 1, 1, 9, 14, 32, 184, 123, 696, 935, 6554, 21105, 60756, 241780, 517970, 1835109, 5741024, 16091004, 63590090, 285113492, 1098219807 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

Theorem: For any integer n > 5, there is a circular permutation i_0, i_1, ..., i_n of 0, 1, ..., n with both 1 and 4 adjacent to 0 such that all the n+1 adjacent distances |i_0-i_1|, |i_1-i_2|, ..., |i_{n-1}-i_n|, |i_n-i_0| are perfect squares.

Proof: By examples, the result holds for n = 6, ..., 11. Below we assume n > 11  and exhibit a circular permutation meeting the requirement.

  If n == 0 (mod 6), then we may take the circular permutation (n,n-4,n-5,n-6,n-10,n-11,n-12,...,8,7,6,2,3,4,0,1,5,9,10,11,...,n-9,n-8,n-7,n-3,n-2,n-1).

  If n == 3 (mod 6), then we may take the circular permutation

  (n,n-4,n-5,n-6,n-10,n-11,n-12,...,11,10,9,5,1,0,4,3,2,6,7,8,...,n-9,n-8,n-7,n-3,n-2,n-1).

  If n == 1 (mod 6), then we may take the circular permutation

  (n,n-4,n-5,n-6,n-10,n-11,n-12,...,9,8,7,3,2,1,0,4,5,6,10,11,12,...,n-9,n-8,n-7,n-3,n-2,n-1).

  If n == 4 (mod 6), then we may take the circular permutation (n,n-4,n-5,n-6,n-10,n-11,n-12,...,12,11,10,6,5,4,0,1,2,3,7,8,9,...,n-9,n-8,n-7,n-3,n-2,n-1).

  If n == 2 (mod 6), then we may take the circular permutation (n,n-4,n-5,n-6,n-10,n-11,n-12,...,10,9,8,4,0,1,5,6,2,3,7,11,12,13,...,n-9,n-8,n-7,n-3,n-2,n-1).

  If n == 5 (mod 6), then we may take the circular permutation

  (n,n-4,n-5,n-6,n-10,n-11,n-12,...,13,12,11,7,3,2,6,5,1,0,4,8,9,10,...,n-9,n-8,n-7,n-3,n-2,n-1).

Zhi-Wei Sun also used a similar method to show that for any positive integer n not equal to 2 or 4 there is a circular permutation i_0, i_1, ..., i_n of 0, 1, ..., n such that all the n+1 adjacent distances |i_0-i_1|, |i_1-i_2|, ..., |i_{n-1}-i_n|, |i_n-i_0| are triangular numbers.

LINKS

Table of n, a(n) for n=1..26.

Zhi-Wei Sun, Some new problems in additive combinatorics, preprint, arXiv:1309.1679 [math.NT], 2013-2014.

EXAMPLE

a(1) = 1 due to the circular permutation (0,1).

a(4) = 1 due to the circular permutation (0,1,2,3,4).

a(6) = 1 due to the circular permutation (0,1,5,6,2,3,4).

a(7) = 1 due to the circular permutation (0,1,2,3,7,6,5,4).

a(8) = 1 due to the circular permutation (0,1,5,6,2,3,7,8,4).

a(9) = 9 due to the circular permutations

  (0,1,2,3,4,5,6,7,8,9), (0,1,2,3,4,8,7,6,5,9),

  (0,1,2,3,7,6,5,4,8,9), (0,1,2,3,7,6,5,9,8,4),

  (0,1,2,6,5,4,3,7,8,9), (0,1,2,6,5,9,8,7,3,4),

  (0,1,5,4,3,2,6,7,8,9), (0,1,5,9,8,7,6,2,3,4),

  (0,4,3,2,1,5,6,7,8,9).

a(10) > 0 due to the permutation (0,1,2,3,7,8,9,10,6,5,4).

a(11) > 0 due to the permutation (0,1,5,9,8,7,11,10,6,2,3,4).

MATHEMATICA

(* A program to compute required circular permutations for n = 8. To get "undirected" circular permutations, we should identify a circular permutation with the one of the opposite direction. Thus a(8) is half of the number of circular permutations yielded by this program. *)

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]

f[i_, j_]:=f[i, j]=SQ[Abs[i-j]]

V[i_]:=V[i]=Part[Permutations[{1, 2, 3, 4, 5, 6, 7, 8}], i]

m=0

Do[Do[If[f[If[j==0, 0, Part[V[i], j]], If[j<8, Part[V[i], j+1], 0]]==False, Goto[aa]], {j, 0, 8}]; m=m+1; Print[m, ":", " ", 0, " ", Part[V[i], 1], " ", Part[V[i], 2], " ", Part[V[i], 3], " ", Part[V[i], 4], " ", Part[V[i], 5], " ", Part[V[i], 6], " ", Part[V[i], 7], " ", Part[V[i], 8]]; Label[aa]; Continue, {i, 1, 8!}]

CROSSREFS

Cf. A000290, A185645, A228626, A228728, A228956, A229005.

Sequence in context: A329751 A093021 A272781 * A175201 A066793 A275508

Adjacent sequences:  A229540 A229541 A229542 * A229544 A229545 A229546

KEYWORD

nonn,more,hard

AUTHOR

Zhi-Wei Sun, Sep 25 2013

EXTENSIONS

a(10)-a(24) from Alois P. Heinz, Sep 26 2013

a(25)-a(26) from Max Alekseyev, Jan 08 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 26 16:29 EDT 2020. Contains 337374 sequences. (Running on oeis4.)