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A253472
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Square Pairs: Numbers n such that 1, 2, ..., 2n can be partitioned into n pairs, where each pair adds up to a perfect square.
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3
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4, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 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
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OFFSET
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1,1
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COMMENTS
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Kiran Kedlaya proved that all numbers greater than 11 are included in the sequence. Outline of proof:
- Show by hand or by computer that it works up to n = 30.
- 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.
- For n=49, ..., 72 pairs adding up to 13^2 allow us to reduce to n=35.
- 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.
- This always works, as long as (2m+1)^2 - 25 > 8m+31, and therefore we must have m > 4.
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.
Worthy of consideration for the elementary school classroom working on square numbers. - Gordon Hamilton, Mar 20 2015
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REFERENCES
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Alfred S. Posamentier, Stephen Krulik, Problem-Solving Strategies for Efficient and Elegant Solutions, Grades 6-12, 2008, page 191.
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LINKS
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Gordon Hamilton, Kiran S. Kedlaya, and Henri Picciotto, Square-Sum Pair Partitions, College Mathematics Journal 46.4 (2015): 264-269.
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FORMULA
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a(n) = 2*a(n-1) - a(n-2) for n > 6.
G.f.: x*(-2*x^5 + 2*x^4 - 2*x^2 - x + 4)/(x - 1)^2. (End)
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EXAMPLE
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For n = 4: (8, 1), (7, 2), (6, 3), (5, 4).
For n = 7: (14, 2), (13, 3), (12, 4), (11, 5), (10, 6), (9, 7), (8, 1).
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PROG
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(Python) # See link.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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