

A090461


Numbers n such that there is a permutation of the numbers 1 to n such that the sum of adjacent numbers is a square.


4



15, 16, 17, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89
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OFFSET

1,1


COMMENTS

Conjecture that there is a solution for all n > 24. See A090460 for a count of the number of essentially different solutions.
It is now known that 25..299 are in the sequence, see the Numberphile 2 link.  Jud McCranie, Jan 11 2018
Every 25 <= n <= 2^20 is in the sequence and (71*25^k1)/2 is also in the sequence for every k, hence this sequence is infinite, see Mersenneforum link for the proof, we give Hamiltonian cycle for these n values if n >= 32.  Robert Gerbicz, Jan 17 2017
The conjecture has been proved: every n >= 25 is in the sequence, moreover for n >= 32 there is a Hamiltonian cycle, see Mersenneforum topic for a code and deterministic algorithm to find a sequence.  Robert Gerbicz, Jan 21 2017


LINKS

Table of n, a(n) for n=1..69.
Brady Haran and Matt Parker, The SquareSum Problem  , Numberphile video (2018)
Brady Haran, Matt Parker, and Charlie Turner, The SquareSum Problem (extra footage)  Numberphile 2 (2018)
Mersenneforum, The SquareSum problem


EXAMPLE

See A071983.


MAPLE

F:= proc(n)
uses GraphTheory;
local edg, G;
edg:= select(t > issqr(t[1]+t[2]), {seq(seq({i, j}, i=1..j1), j=1..n)}) union {seq({i, n+1}, i=1..n)};
G:= Graph(n+1, edg);
IsHamiltonian(G)
end proc:
select(F, [$1..50]); # Robert Israel, Jun 05 2015


CROSSREFS

Cf. A071983, A071984 (number of circular solutions), A090460.
Cf. A078107 (n for which there is no solution).
Sequence in context: A297281 A176294 A214424 * A138598 A281879 A160661
Adjacent sequences: A090458 A090459 A090460 * A090462 A090463 A090464


KEYWORD

nonn,changed


AUTHOR

T. D. Noe, Dec 01 2003


EXTENSIONS

a(31)a(69) from Donovan Johnson, Sep 14 2010


STATUS

approved



