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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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5
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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]
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REFERENCES
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Lehmer, D. H.; 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]
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LINKS
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Table of n, a(n) for n=1..42.
Index entries for sequences related to sums of squares
Index to sequences with linear recurrences with constant coefficients, signature (0,0,4).
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FORMULA
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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
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MATHEMATICA
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Select[Range[0, 3584], Length[PowersRepresentations[ #, 4, 2]] == 1&] (* Ant King, Aug 15 2010 *)
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PROG
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(PARI) {a(n)=if(n<1, 0, if(n<14, [1, 2, 3, 5, 6, 7, 8, 11, 14, 15, 23, 24, 32] [n], [4, 7, 12][(n+1)%3+1]*2^((n+1)\3*2-7)))} /* Michael Somos Apr 08 2006 */
(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 */
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CROSSREFS
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Cf. A180149.
Sequence in context: A191893 A016741 A191167 * A151894 A028229 A104452
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
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STATUS
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approved
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