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

0,1

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

References

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

Links

Table of n, a(n) for n=0..40.

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

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

Example

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.

Mathematica

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

Prog

(Magma) [4*n^2 + 72*n + 454: n in [0..50]] /* or */ 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
(PARI) a(n)=4*n^2+72*n+454 \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

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

Sequence in context: A245946 A076547 A365619 * A213532 A234771 A297658

Adjacent sequences: A259055 A259056 A259057 * A259059 A259060 A259061

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 12 2015

Status

approved