

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



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

1,1


COMMENTS

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)^225, aiming for sums of (2m+3)^2. The first such pair is (2m+1)^223, 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 2n1. All numbers greater than 6 are included.
Worthy of consideration for the elementary school classroom working on square numbers.  Gordon Hamilton, Mar 20 2015


REFERENCES

Alfred S. Posamentier, Stephen Krulik, ProblemSolving Strategies for Efficient and Elegant Solutions, Grades 612, 2008, page 191.


LINKS

Table of n, a(n) for n=1..73.
Gordon Hamilton, Kiran S. Kedlaya, and Henri Picciotto, SquareSum Pair Partitions, College Mathematics Journal 46.4 (2015): 264269.
Thomas Kilkelly, The ARML Power Contest, 2015, page 77.
Henri Picciotto, Python program to generate sequence
Index entries for linear recurrences with constant coefficients, signature (2,1).


FORMULA

From Chai Wah Wu, Aug 13 2020: (Start)
a(n) = 2*a(n1)  a(n2) for n > 6.
G.f.: x*(2*x^5 + 2*x^4  2*x^2  x + 4)/(x  1)^2. (End)


EXAMPLE

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).


PROG

(Python) # See link.


CROSSREFS

Cf. A252897.
Sequence in context: A161986 A324940 A020670 * A255060 A047538 A074231
Adjacent sequences: A253469 A253470 A253471 * A253473 A253474 A253475


KEYWORD

nonn


AUTHOR

Henri Picciotto, Jan 01 2015


STATUS

approved



