

A006431


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


8



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
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OFFSET

1,3


COMMENTS

From a(16) = 96 onwards, the terms of this sequence satisfy the third order recurrence relation a(n) = 4a(n3). [Ant King, Aug 15 2010]
A002635(a(n)) = 1.  Reinhard Zumkeller, Jul 13 2014


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. 476481. [Ant King, Aug 15 2010]
Index entries for sequences related to sums of squares
Index to sequences with linear recurrences with constant coefficients, signature (0,0,4).


FORMULA

Consists of 7 odd numbers plus 0 and numbers of forms 2*4^k, 6*4^k, 14*4^k, k >= 0.
{n  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


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 *)


PROG

(PARI) {a(n)=if(n<2, 0, if(n<15, [1, 2, 3, 5, 6, 7, 8, 11, 14, 15, 23, 24, 32] [n1], [4, 7, 12][n%3+1]*2^(n\3*27)))} /* Michael Somos, Apr 23 2006 */
(Haskell)
a006431 n = a006431_list !! (n1)
a006431_list = filter ((== 1) . a002635) [0..]
 Reinhard Zumkeller, Jul 13 2014


CROSSREFS

Cf. A002635, A180149, A245022.
Sequence in context: A191893 A016741 A191167 * A151894 A028229 A104452
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
A pari/gp script removed by Michel Marcus, Oct 15 2013


STATUS

approved



