This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A259058 #31 Mar 18 2025 21:12:16
%S A259058 454,530,614,706,806,914,1030,1154,1286,1426,1574,1730,1894,2066,2246,
%T A259058 2434,2630,2834,3046,3266,3494,3730,3974,4226,4486,4754,5030,5314,
%U A259058 5606,5906,6214,6530,6854,7186,7526,7874,8230,8594,8966,9346,9734
%N A259058 Numbers that are representable in at least two ways as sums of four distinct nonvanishing squares.
%C A259058 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.
%C A259058 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.
%C A259058 There may be other numbers having this property.
%C A259058 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
%D A259058 W. Sierpiński, 250 Problems in Elementary Number Theory, American Elsevier Publ. Comp., New York, PWN-Polish Scientific Publishers, Warszawa, 1970.
%H A259058 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A259058 a(n) = 4*n^2 + 72*n + 454 = 2*A259059(n). See the comment for the sum of four squares in two ways.
%F A259058 O.g.f.: 2*(227 - 416*x + 193*x^2)/(1-x)^3.
%F A259058 a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - _Vincenzo Librandi_, Aug 13 2015
%e A259058 n=0: 454 = 1^2 + 8^2 + 10^2 + 17^2 = 2^2 + 5^2 + 13^2 + 16^2.
%e A259058 n=2: 614 = 3^2 + 10^2 + 12^2 + 19^2 = 4^2 + 7^2 + 15^2 + 18^2.
%t A259058 CoefficientList[Series[2 (227 - 416 x + 193 x^2)/(1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 13 2015 *)
%o A259058 (Magma) [4*n^2 + 72*n + 454: n in [0..50]]; // _Vincenzo Librandi_, Aug 13 2015
%o A259058 (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
%o A259058 (PARI) a(n)=4*n^2+72*n+454 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A259058 Cf. A259059, A223727, A004433, A259060 (four cubes).
%K A259058 nonn,easy
%O A259058 0,1
%A A259058 _Wolfdieter Lang_, Aug 12 2015