%I
%S 4,7,8,9,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,
%T 32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,
%U 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
%N Square Pairs: Numbers n such that 1, 2, ..., 2n can be partitioned into n pairs, where each pair adds up to a perfect square.
%C Kiran Kedlaya proved that all numbers greater than 11 are included in the sequence. Outline of proof:
%C  Show by hand or by computer that it works up to n = 30.
%C  For n=31, pair 62+59=61+60=11^2 and then reduce to the case of n=29. For n=32, pair 64+57, ..., 61+60 and reduce to the case of 28. And so on. This works until n=48.
%C  For n=49, ..., 72 pairs adding up to 13^2 allow us to reduce to n=35.
%C  Repeat the process, always terminating at (2m+1)^225, aiming for sums of (2m+3)^2. The first such pair is (2m+1)^223, 8m+31.
%C  This always works, as long as (2m+1)^2  25 > 8m+31, and therefore we must have m > 4.
%C A similar sequence using odd numbers can be created, by making n pairs that sum to perfect squares, using numbers from 0 to 2n1. All numbers greater than 6 are included.
%C Worthy of consideration for the elementary school classroom working on square numbers.  _Gordon Hamilton_, Mar 20 2015
%D Alfred S. Posamentier, Stephen Krulik, ProblemSolving Strategies for Efficient and Elegant Solutions, Grades 612, 2008, page 191.
%H Gordon Hamilton, Kiran S. Kedlaya, and Henri Picciotto, <a href="http://www.jstor.org/stable/10.4169/college.math.j.46.4.264">SquareSum Pair Partitions</a>, College Mathematics Journal 46.4 (2015): 264269.
%H Thomas Kilkelly, <a href="http://bookstore.ams.org/mcl15">The ARML Power Contest</a>, 2015, page 77.
%H Henri Picciotto, <a href="/A253472/a253472.py.txt">Python program to generate sequence</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,1).
%F From _Chai Wah Wu_, Aug 13 2020: (Start)
%F a(n) = 2*a(n1)  a(n2) for n > 6.
%F G.f.: x*(2*x^5 + 2*x^4  2*x^2  x + 4)/(x  1)^2. (End)
%e For n = 4: (8, 1), (7, 2), (6, 3), (5, 4).
%e For n = 7: (14, 2), (13, 3), (12, 4), (11, 5), (10, 6), (9, 7), (8, 1).
%o (Python) # See link.
%Y Cf. A252897.
%K nonn
%O 1,1
%A _Henri Picciotto_, Jan 01 2015
