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A006431 Numbers that have a unique partition into a sum of four nonnegative squares. 26
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 (list; graph; refs; listen; history; text; internal format)
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
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.
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
Sequence in context: A191893 A016741 A191167 * A285528 A151894 A352328
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

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Last modified July 24 08:28 EDT 2024. Contains 374575 sequences. (Running on oeis4.)