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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
6, 3, 3, 72, 226, 358, 309, 391, 547, 813, 3562, 10741, 36633, 94547, 120424, 393670, 676579 (list; graph; refs; listen; history; text; internal format)
OFFSET

31,1

LINKS

Table of n, a(n) for n=31..47.

EXAMPLE

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.).

CROSSREFS

Cf. A108658 = square chains.

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

Sequence in context: A153841 A021616 A019164 * A117042 A227989 A189088

Adjacent sequences:  A108658 A108659 A108660 * A108662 A108663 A108664

KEYWORD

hard,nice,nonn,more

AUTHOR

Zak Seidov, T. D. Noe & Max Alekseyev Jun 16 2005

EXTENSIONS

a(42)-a(47) from Donovan Johnson, Sep 14 2010

STATUS

approved

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Last modified November 27 17:39 EST 2014. Contains 250249 sequences.