%I #53 May 08 2024 02:27:07
%S 1,1,11,57,31,20,25,50,64,464,1062,4337,10091,21931,69623,115913,
%T 227893,457707,297126,1051583,3377189,7618873,12476654,25832098,
%U 55792448,75126741,129180538,357114149,823402071,3902161448,20043267339,131420398568,347422743997,811591067418
%N Square loops: the number of circular permutations (reversals not counted as different) of the numbers 1 to n such that the sum of any two consecutive numbers is a square.
%C It is unknown whether a circular permutation of the numbers 1 to n exists such that the sum of any two consecutive numbers is a cube.
%C According to Rivera's Puzzle 311, the smallest n for which a cubic loop exists is 473. - _T. D. Noe_, Nov 26 2007
%C From _Bert Dobbelaere_, Dec 28 2018: (Start)
%C It is easy to see that no solutions for n <= 30 can exist: for each value of n <= 30 at least one number exists that can only be paired with at most one other number to form a square (e.g., 18 for n=30 can only be paired with 7). No Hamiltonian cycle can exist if the graph contains a vertex of degree less than 2.
%C For the case n=31, the nonexistence of a Hamiltonian cycle is less trivial but can be shown by hand.
%C (End)
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_311.htm">Puzzle 311: Sum to a cube</a>, The Prime Puzzles and Problems Connection.
%F a(n) = (A071983(n) - A090460(n)) / (n-1) for n > 1. - _Martin Ehrenstein_, May 22 2023
%e There is only one possible square loop of minimum length, which is (32, 4, 21, 28, 8, 1, 15, 10, 26, 23, 2, 14, 22, 27, 9, 16, 20, 29, 7, 18, 31, 5, 11, 25, 24, 12, 13, 3, 6, 30, 19, 17) so a(32)=1.
%Y Cf. A071983, A090460, A112663.
%K nice,nonn,more,hard
%O 32,3
%A _William Rex Marshall_, Jun 16 2002
%E a(48)-a(49) from _Donovan Johnson_, Sep 14 2010
%E a(50)-a(52) from _Giovanni Resta_, Nov 11 2012
%E a(53)-a(54) from _Fausto A. C. Cariboni_, Sep 19 2018
%E a(55) from _Jud McCranie_, Sep 30 2018
%E a(56) from _Jud McCranie_, Oct 08 2018
%E a(57) from _Fausto A. C. Cariboni_, Oct 24 2018
%E a(58)-a(61) from _Bert Dobbelaere_, Dec 28 2018
%E a(62)-a(63) from _Martin Ehrenstein_, May 22 2023
%E a(64) from _Zhao Hui Du_, Apr 30 2024
%E a(65) from _Zhao Hui Du_, May 08 2024