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A006431
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Numbers that have a unique partition into a sum of four nonnegative squares.
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26
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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
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1,3
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COMMENTS
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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
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REFERENCES
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E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.
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LINKS
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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. [Ant King, Aug 15 2010]
Index entries for sequences related to sums of squares
Index entries for linear recurrences with constant coefficients, signature (0,0,4).
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A002635, A180149, A245022.
Sequence in context: A191893 A016741 A191167 * A285528 A151894 A028229
Adjacent sequences: A006428 A006429 A006430 * A006432 A006433 A006434
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KEYWORD
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nonn,easy,nice
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AUTHOR
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David M. Bloom.
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EXTENSIONS
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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
Edited, Grosswald reference added. - Wolfdieter Lang, Aug 12 2015
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STATUS
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approved
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