A257542 Square-sum pairs: Numbers n such that 0,1, ..., 2n-1 can be partitioned into n pairs, where each pair adds up to a perfect square.

1, 4, 5, 7, 8, 9, 10, 11, 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

Offset: 1

Author

Brian Hopkins, Apr 28 2015

Keywords

• nonn
• easy

Comments

Kilkelly uses induction to prove that all integers greater than 20 are in the sequence after using various methods on smaller cases.

The positive integers except 2, 3, and 6.

The positive integers except the strong divisors of 6. - Omar E. Pol, Apr 30 2015

Examples

			For n = 4: (0, 1), (2, 7), (3, 6), (4, 5)
For n = 7: (0, 9), (1, 8), (2, 7), (3, 13), (4, 12), (5, 11), (6, 10)
		

References

• T. Kilkelly, The ARML Power Contest, American Mathematical Society, 2015, chapter 11.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,-1).

Crossrefs

Cf. A253472, A252897.

Essentially the same as A055495.

Programs

PARI

is(n)=n>6 || n==1 || n==4 || n==5 \\ Charles R Greathouse IV, Apr 30 2015

Formula

From Chai Wah Wu, Aug 13 2020: (Start)

a(n) = 2*a(n-1) - a(n-2) for n > 5.

G.f.: x*(-x^4 + x^3 - 2*x^2 + 2*x + 1)/(x - 1)^2. (End)