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A253472 Square Pairs: Numbers n such that 1, 2, ..., 2n can be partitioned into n pairs, where each pair adds up to a perfect square. 3

%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)^2-25, aiming for sums of (2m+3)^2. The first such pair is (2m+1)^2-23, 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 2n-1. 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, Problem-Solving Strategies for Efficient and Elegant Solutions, Grades 6-12, 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">Square-Sum Pair Partitions</a>, College Mathematics Journal 46.4 (2015): 264-269.

%H Thomas Kilkelly, <a href="http://bookstore.ams.org/mcl-15">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(n-1) - a(n-2) 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

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Last modified November 24 18:11 EST 2020. Contains 338616 sequences. (Running on oeis4.)