A259058 Numbers that are representable in at least two ways as sums of four distinct nonvanishing squares.

454, 530, 614, 706, 806, 914, 1030, 1154, 1286, 1426, 1574, 1730, 1894, 2066, 2246, 2434, 2630, 2834, 3046, 3266, 3494, 3730, 3974, 4226, 4486, 4754, 5030, 5314, 5606, 5906, 6214, 6530, 6854, 7186, 7526, 7874, 8230, 8594, 8966, 9346, 9734

Offset: 0

Author

Wolfdieter Lang, Aug 12 2015

Keywords

• nonn
• easy

Comments

This is part one of Exercise 229 in Sierpiński's problem book. See p. 20 and p. 110 for the solution. He uses the identity (n-8)^2 + (n-1)^2 + (n+1)^2 + (n+8)^2 = 4*n^2 + 130 = (n-7)^2 + (n-4)^2 + (n+4)^2 + (n+7)^2, for n >= 9.

Here n was replaced by n + 9: (n+1)^2 + (n+8)^2 +(n+10)^2 + (n+17)^2 = 4*n^2 + 72*n + 454 = (n+2)^2 + (n+5)^2 + (n+13)^2 + (n+16)^2, for n >= 0.

There may be other numbers having this property.

Because the summands have no common factor > 1 each of these two representations is called primitive. Therefore, this is a proper subsequence of A223727, hence of A004433. - Wolfdieter Lang, Aug 20 2015

Examples

			n=0: 454 = 1^2 + 8^2 + 10^2 + 17^2 = 2^2 + 5^2 + 13^2 + 16^2.
n=2: 614 = 3^2 + 10^2 + 12^2 + 19^2 = 4^2 + 7^2 + 15^2 + 18^2.
		

References

• W. Sierpiński, 250 Problems in Elementary Number Theory, American Elsevier Publ. Comp., New York, PWN-Polish Scientific Publishers, Warszawa, 1970.

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A259059, A223727, A004433, A259060 (four cubes).

Programs

Magma

[4*n^2 + 72*n + 454: n in [0..50]]; // Vincenzo Librandi, Aug 13 2015

Magma

I:=[454, 530, 614]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 13 2015

Mathematica

CoefficientList[Series[2 (227 - 416 x + 193 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 13 2015 *)

PARI

a(n)=4*n^2+72*n+454 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 4*n^2 + 72*n + 454 = 2*A259059(n). See the comment for the sum of four squares in two ways.

O.g.f.: 2*(227 - 416*x + 193*x^2)/(1-x)^3.

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Vincenzo Librandi, Aug 13 2015