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A108661 Square loops: the number of circular permutations (reversals not counted as different) of the numbers 0 to n such that the sum of any two consecutive numbers is a square. 1

%I

%S 6,3,3,72,226,358,309,391,547,813,3562,10741,36633,94547,120424,

%T 393670,676579,1088429,5531195,3294327,8335128,27820643,75288569,

%U 111875702,264015370,465407197,687532936,1109951444,3256360099

%N Square loops: the number of circular permutations (reversals not counted as different) of the numbers 0 to n such that the sum of any two consecutive numbers is a square.

%e There is no solution for n=0,...,30, (note offset=31). For n=0,1 we have the trivial square circles {0} and {0,1}, which are not included in the sequence.

%e There are only six possible square loops of the minimum length (n=31 case):

%e {1,0,4,5,31,18,7,29,20,16,9,27,22,3,13,12,24,25,11,14,2,23,26,10,6,30,19,17,8,28,21,15},

%e {1,0,4,12,13,3,6,30,19,17,8,28,21,15,10,26,23,2,14,22,27,9,16,20,29,7,18,31,5,11,25,24},

%e {1,0,4,21,28,8,17,19,30,6,3,13,12,24,25,11,5,31,18,7,29,20,16,9,27,22,14,2,23,26,10,15},

%e {1,15,10,26,23,2,14,22,27,9,16,20,29,7,18,31,5,11,25,0,4,21,28,8,17,19,30,6,3,13,12,24},

%e {1,15,21,28,8,17,19,30,6,10,26,23,2,14,11,5,31,18,7,29,20,16,9,27,22,3,13,12,4,0,25,24},

%e {1,15,21,28,8,17,19,30,6,10,26,23,2,14,11,25,0,4,5,31,18,7,29,20,16,9,27,22,3,13,12,24}.

%e In the n=32,33 (resp.) cases, there are three square loop solutions:

%e {1,0,4,32,17,19,30,6,3,13,12,24,25,11,5,31,18,7,29,20,16,9,27,22,14,2,23,26,10,15,21,28,8},

%e {1,8,28,21,4,32,17,19,30,6,3,13,12,24,25,11,5,31,18,7,29,20,16,0,9,27,22,14,2,23,26,10,15},

%e {1,8,28,21,15,10,26,23,2,14,22,27,9,16,20,29,7,18,31,5,11,25,0,4,32,17,19,30,6,3,13,12,24},

%e and

%e {1,0,4,32,17,19,30,6,3,13,12,24,25,11,5,20,29,7,18,31,33,16,9,27,22,14,2,23,26,10,15,21,28,8},

%e {1,8,28,21,4,32,17,19,30,6,3,13,12,24,25,11,5,20,29,7,18,31,33,16,0,9,27,22,14,2,23,26,10,15},

%e {1,8,28,21,15,10,26,23,2,14,22,27,9,16,33,31,18,7,29,20,5,11,25,0,4,32,17,19,30,6,3,13,12,24}

%e (resp.).

%Y Cf. A108658 = square chains.

%Y Cf. A071984, A090460, A108658, A108659, A108660.

%K hard,nice,nonn,more

%O 31,1

%A _Zak Seidov_, _T. D. Noe_ & _Max Alekseyev_ Jun 16 2005

%E a(42)-a(47) from _Donovan Johnson_, Sep 14 2010

%E a(48)-a(52) from _Fausto A. C. Cariboni_, Sep 21 2018

%E a(53)-a(59) from _Bert Dobbelaere_, Dec 29 2018

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Last modified June 13 20:01 EDT 2021. Contains 345009 sequences. (Running on oeis4.)