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

1,3

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).

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

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 *)

Prog

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

Crossrefs

Cf. A002635, A180149, A245022.

Sequence in context: A191893 A016741 A191167 * A285528 A151894 A352328

Adjacent sequences: A006428 A006429 A006430 * A006432 A006433 A006434

Keyword

nonn,easy,nice

Author

David M. Bloom

Extensions

More terms from James A. 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

Status

approved