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 A006431 #72 Jul 02 2025 16:01:54
%S A006431 0,1,2,3,5,6,7,8,11,14,15,23,24,32,56,96,128,224,384,512,896,1536,
%T A006431 2048,3584,6144,8192,14336,24576,32768,57344,98304,131072,229376,
%U A006431 393216,524288,917504,1572864,2097152,3670016,6291456,8388608,14680064
%N A006431 Numbers that have a unique partition into a sum of four nonnegative squares.
%C A006431 From a(16) = 96 onwards, the terms of this sequence satisfy the third-order recurrence relation a(n) = 4a(n-3). - _Ant King_, Aug 15 2010
%C A006431 A002635(a(n)) = 1. - _Reinhard Zumkeller_, Jul 13 2014
%D A006431 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.
%H A006431 Vincenzo Librandi, <a href="/A006431/b006431.txt">Table of n, a(n) for n = 1..1000</a>
%H A006431 Pierre de la Harpe, <a href="http://images-archive.math.cnrs.fr/Lagrange-et-la-variation-des.html">Lagrange et la variation des théorèmes</a>, Images des Mathématiques, CNRS, 2014.
%H A006431 D. H. Lehmer, <a href="http://www.jstor.org/stable/2305380">On the Partition of Numbers into Squares</a>, The American Mathematical Monthly, Vol. 55, No.8, October 1948, pp. 476-481.
%H A006431 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%H A006431 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,4).
%F A006431 Consists of the seven odd numbers 1, 3, 5, 7, 11, 15, 23, plus 0, and numbers of forms 2*4^k, 6*4^k, 14*4^k, k >= 0.
%F A006431 The set {n nonnegative : A002635(n) = 1}.
%F A006431 G.f.: x^2*(36*x^13 +28*x^12 +32*x^11 +21*x^10 +17*x^9 +14*x^8 +13*x^7 +12*x^6 +5*x^5 +2*x^4 -x^3 -3*x^2 -2*x -1) / (4*x^3 -1). - _Colin Barker_, Apr 20 2013
%F A006431 log(a(n)) = n*log(4)/3 + C(n) + o(1) where C(n) ~ {-2.82922, -3.00364, -2.90612} for n (mod 3) == {2,0,1}. - _Bill McEachen_, Oct 21 2022
%t A006431 Select[Range[0,3584], Length[PowersRepresentations[ #,4,2]] == 1&] (* _Ant King_, Aug 15 2010 *)
%t A006431 CoefficientList[Series[x (36 x^13 + 28 x^12 + 32 x^11 + 21 x^10 + 17 x^9 + 14 x^8 + 13 x^7 + 12 x^6 + 5 x^5 + 2 x^4 - x^3 - 3 x^2 - 2 x - 1)/(4 x^3 - 1), {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 14 2013 *)
%t A006431 LinearRecurrence[{0,0,4},{0,1,2,3,5,6,7,8,11,14,15,23,24,32,56},50] (* _Harvey P. Dale_, Nov 26 2015 *)
%o A006431 (PARI) {a(n)=if(n<2, 0, if(n<15, [1, 2, 3, 5, 6, 7, 8, 11, 14, 15, 23, 24, 32] [n-1], [4, 7, 12][n%3+1]*2^(n\3*2-7)))} /* _Michael Somos_, Apr 23 2006 */
%o A006431 (Haskell)
%o A006431 a006431 n = a006431_list !! (n-1)
%o A006431 a006431_list = filter ((== 1) . a002635) [0..]
%o A006431 -- _Reinhard Zumkeller_, Jul 13 2014
%Y A006431 Cf. A002635, A180149, A245022.
%K A006431 nonn,easy,nice
%O A006431 1,3
%A A006431 David M. Bloom
%E A006431 More terms from _James Sellers_, Dec 24 1999
%E A006431 Corrected by _T. D. Noe_, Jun 15 2006
%E A006431 Definition revised by _Ant King_, May 06 2010
%E A006431 Edited and Grosswald reference added by _Wolfdieter Lang_, Aug 12 2015