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A002635
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Number of partitions of n into 4 squares.
(Formerly M0053 N0018)
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24
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1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 3, 2, 2, 2, 2, 1, 1, 3, 3, 3, 3, 2, 2, 2, 1, 3, 4, 2, 4, 3, 3, 2, 2, 3, 4, 3, 2, 4, 2, 2, 2, 4, 5, 3, 5, 3, 5, 3, 1, 4, 5, 3, 3, 4, 3, 4, 2, 4, 6, 4, 4, 4, 5, 2, 3, 5, 5, 5, 5, 4, 4, 3, 2, 6, 7, 4, 5, 5, 5, 4, 2, 5, 9, 5, 3, 5, 4, 3, 1, 6, 7, 6, 7, 5, 7, 5, 3, 6, 7, 4
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OFFSET
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0,5
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COMMENTS
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a(A124978(n)) = n; a(A006431(n)) = 1; a(A180149(n)) = 2; a(A245022(n)) = 3. - Reinhard Zumkeller, Jul 13 2014
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REFERENCES
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G. Loria, Sulla scomposizione di un intero nella somma di numeri poligonali. (Italian) Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1, (1946). 7-15.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..10000
E. Grosswald, The Problem of the Uniqueness of Essentially Distinct Representations, in Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 84.
D. H. Lehmer, Review of Loria article, Math. Comp. 2 (1947), 301-302.
Gino Loria, Sulla scomposizione di un intero nella somma di numeri poligonali. (Italian). Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 1, (1946). 7-15. Also D. H. Lehmer, Review of Loria article, Math. Comp. 2 (1947), 301-302. [Annotated scanned copies]
M. D. Hirschhorn, Some formulas for partitions into squares, Discrete Math, 211 (2000), pp. 225-228.
James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.
Index entries for sequences related to sums of squares
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EXAMPLE
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1: 1000; 2: 1100; 3:1110; 4: 2000 and 1111; 5: 2100; 6: 2110; 7: 2111; 8: 2200; 9: 3000 and 2210; 10: 3100 and 2211; etc.
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MATHEMATICA
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Length[PowersRepresentations[ #, 4, 2]] & /@ Range[0, 107] (* Ant King, Oct 19 2010 *)
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PROG
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(PARI) for(n=1, 100, print1(sum(a=0, n, sum(b=0, a, sum(c=0, b, sum(d=0, c, if(a^2+b^2+c^2+d^2-n, 0, 1))))), ", "))
(PARI) a(n)=local(c=0); if(n>=0, forvec(x=vector(4, k, [0, sqrtint(n)]), c+=norml2(x)==n, 1)); c
(Haskell)
a002635 = p (tail a000290_list) 4 where
p ks'@(k:ks) c m = if m == 0 then 1 else
if c == 0 || m < k then 0 else p ks' (c - 1) (m - k) + p ks c m
-- Reinhard Zumkeller, Jul 13 2014
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CROSSREFS
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Cf. A000290, A124978, A006431, A180149, A245022.
Equivalent sequences for other numbers of squares: A010052 (1), A000161 (2), A000164 (3), A000174 (5), A000177 (6), A025422 (7), A025423 (8), A025424 (9), A025425 (10).
Sequence in context: A033182 A053797 A254011 * A275806 A228369 A296773
Adjacent sequences: A002632 A002633 A002634 * A002636 A002637 A002638
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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