A006431 Numbers that have a unique partition into a sum of four nonnegative squares.

0, 1, 2, 3, 5, 6, 7, 8, 11, 14, 15, 23, 24, 32, 56, 96, 128, 224, 384, 512, 896, 1536, 2048, 3584, 6144, 8192, 14336, 24576, 32768, 57344, 98304, 131072, 229376, 393216, 524288, 917504, 1572864, 2097152, 3670016, 6291456, 8388608, 14680064

Offset: 1

Author

David M. Bloom

Keywords

• nonn
• easy
• nice

Comments

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

A002635(a(n)) = 1. - Reinhard Zumkeller, Jul 13 2014

References

• E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Pierre de la Harpe, Lagrange et la variation des théorèmes, Images des Mathématiques, CNRS, 2014.
• D. H. Lehmer, On the Partition of Numbers into Squares, The American Mathematical Monthly, Vol. 55, No.8, October 1948, pp. 476-481.
• Index entries for sequences related to sums of squares
• Index entries for linear recurrences with constant coefficients, signature (0,0,4).

Crossrefs

Cf. A002635, A180149, A245022.

Programs

Haskell

a006431 n = a006431_list !! (n-1)
a006431_list = filter ((== 1) . a002635) [0..]
-- Reinhard Zumkeller, Jul 13 2014

Mathematica

Select[Range[0,3584], Length[PowersRepresentations[ #,4,2]] == 1&] (* Ant King, Aug 15 2010 *)
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 *)
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 *)

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 */

Formula

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.

The set {n nonnegative : A002635(n) = 1}.

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

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

Extensions

More terms from James Sellers, Dec 24 1999

Corrected by T. D. Noe, Jun 15 2006

Definition revised by Ant King, May 06 2010

Edited and Grosswald reference added by Wolfdieter Lang, Aug 12 2015