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.
6, 3, 3, 72, 226, 358, 309, 391, 547, 813, 3562, 10741, 36633, 94547, 120424, 393670, 676579, 1088429, 5531195, 3294327, 8335128, 27820643, 75288569, 111875702, 264015370, 465407197, 687532936, 1109951444, 3256360099
Offset: 31
Examples
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.
There are only six possible square loops of the minimum length (n=31 case):
{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},
{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},
{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},
{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},
{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},
{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}.
In the n=32,33 (resp.) cases, there are three square loop solutions:
{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},
{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},
{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},
and
{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},
{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},
{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}
(resp.).
Extensions
a(42)-a(47) from Donovan Johnson, Sep 14 2010
a(48)-a(52) from Fausto A. C. Cariboni, Sep 21 2018
a(53)-a(59) from Bert Dobbelaere, Dec 29 2018