

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.


6



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
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OFFSET

15,9


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


REFERENCES

Ruemmler, Ronald E., "Square Loops," Journal of Recreational Mathematics 14:2 (198182), page 141; Solution by Chris Crandell and Lance Gay, JRM 15:2 (198283), page 155.


LINKS

Table of n, a(n) for n=15..52.
Carlos Rivera, Puzzle 311: Sum to a cube


EXAMPLE

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.


CROSSREFS

Cf. A071984.
Cf. A090460, A090461.
Cf. A078107 (n for which there is no solution).
Sequence in context: A319202 A095200 A090460 * A302693 A215514 A215347
Adjacent sequences: A071980 A071981 A071982 * A071984 A071985 A071986


KEYWORD

more,nice,nonn


AUTHOR

William Rex Marshall, Jun 16 2002


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


STATUS

approved



