A071983 Square chains: the number of permutations (reversals not counted as different) of the numbers 1 to n such that the sum of any two consecutive numbers is a square.

1, 1, 1, 0, 0, 0, 0, 0, 3, 0, 10, 12, 35, 52, 19, 20, 349, 392, 669, 4041, 17175, 12960, 14026, 11889, 29123, 39550, 219968, 553694, 2178103, 5301127, 12220138, 38838893, 68361609, 140571720, 280217025, 204853870, 738704986, 2368147377, 5511090791, 9802605881, 21164463050, 47746712739, 68092497615, 123092214818

Offset: 15

Author

William Rex Marshall, Jun 16 2002

Keywords

• nice
• hard
• nonn

Comments

For n > 31, this sequence counts each circular solution (in which the first and last numbers also sum to a square) n times. Sequence A090460 counts the circular solutions only once, giving the number of essentially different solutions.

The existence of cubic chains in answered affirmatively in Puzzle 311. - T. D. Noe, Jun 16 2005

Examples

			There is only one possible square chain of minimum length, which is: (8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9) so a(15)=1.
		

References

• Ruemmler, Ronald E., "Square Loops," Journal of Recreational Mathematics 14:2 (1981-82), page 141; Solution by Chris Crandell and Lance Gay, JRM 15:2 (1982-83), page 155.

Links

• Zhao Hui Du, Table of n, a(n) for n = 15..59
• Carlos Rivera, Puzzle 311: Sum to a cube, The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A071984.

Cf. A090460, A090461.

Cf. A078107 (n for which there is no solution).

Formula

a(n) = A090460(n) + (n-1)*A071984(n). - Martin Ehrenstein, May 16 2023

Extensions

a(43)-a(45) from Donovan Johnson, Sep 14 2010

a(46)-a(47) from Jud McCranie, Aug 18 2018

a(48) from Jud McCranie, Sep 17 2018

a(49)-a(52) from Bert Dobbelaere, Dec 30 2018

a(53)-a(54) from Martin Ehrenstein, May 16 2023

a(55)-a(56) from Zhao Hui Du, Apr 25 2024

a(57)-a(58) from Zhao Hui Du, Apr 26 2024