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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A274877 A091014 A097370 * A174904 A074390 A255617
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

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Last modified April 22 22:30 EDT 2024. Contains 371906 sequences. (Running on oeis4.)