A252897 Rainbow Squares: a(n) = number of ways to pair the integers 1 to 2n so that the sum of each pair is a square.

1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 6, 18, 12, 36, 156, 295, 429, 755, 2603, 7122, 19232, 32818, 54363, 172374, 384053, 933748, 1639656, 4366714, 20557751, 83801506, 188552665, 399677820, 640628927, 2175071240, 8876685569, 32786873829, 108039828494

Offset

0,14

Comments

The original sequence is from Henri Picciotto who asked for which n is such a pairing possible: A253472.

The name "rainbow squares" refers to the use of this problem in the elementary school classroom where children draw colored connecting "rainbows" to make the pairings.

Number of perfect matchings in the graph with vertices 1 to 2n and edges {i,j} where i+j is a square. - Robert Israel, Mar 22 2015

Links

Table of n, a(n) for n=0..39.

Gordon Hamilton, Kiran S. Kedlaya, and Henri Picciotto, Square-Sum Pair Partitions, The College Mathematics Journal, Vol. 46, No. 4 (September 2015), pp. 264-269.

Example

One of the solutions for n=13 consists of the following pairings of 1-26:
{1,15}, adding to 16;
{2,23}, {3,22}, {4,21}, {5,20}, {6,19}, {7,18}, {8,17}, {9,16}, {11,14}, {12, 13}, each adding to 25;
{10,26}, adding to 36;
{24,25}, adding to 49.
There are five other such pairings possible, so a(13) = 6.

Maple

F:= proc(S)
  option remember;
  local s, ts;
  if nops(S) = 0 then return 1 fi;
  s:= S[-1];
  ts:= select(t -> issqr(s+t), S minus {s});
  add(procname(S minus {s, t}), t = ts);
end proc:
seq(F({$1..2*n}), n = 0 .. 24); # Robert Israel, Mar 22 2015

Mathematica

F[S_] := F[S] = Module[{s, ts}, If[Length[S] == 0, Return[1]]; s = S[[-1]]; ts = Select[S ~Complement~ {s}, IntegerQ[Sqrt[s + #]]&]; Sum[F[S ~Complement~ {s, t}], {t, ts}]];
Table[Print[n]; F[Range[2 n]], {n, 0, 24}] (* Jean-François Alcover, Mar 19 2019, after Robert Israel *)

Crossrefs

Cf. A253472, A278329, A278339.

Sequence in context: A274877 A091014 A097370 * A174904 A074390 A255617

Adjacent sequences: A252894 A252895 A252896 * A252898 A252899 A252900

Keyword

nonn,more

Author

Gordon Hamilton, Mar 22 2015

Extensions

a(26)-a(30) from Hiroaki Yamanouchi, Mar 25 2015

a(31) from Alois P. Heinz, Nov 16 2016

a(32)-a(36) from Linus and Joost VandeVondele, Jun 07 2018

a(37)-a(39) from Bert Dobbelaere, Aug 09 2022

Status

approved