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A002635 Number of partitions of n into 4 squares.
(Formerly M0053 N0018)
9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

a(A124978(n)) = n;  a(A006431(n)) = 1; a(A180149(n)) = 2; a(A245022(n)) = 3. - Reinhard Zumkeller, Jul 13 2014

REFERENCES

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).

LINKS

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. [From Ant King, Oct 19 2010]

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

EXAMPLE

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.

MATHEMATICA

Length[PowersRepresentations[ #, 4, 2]] & /@ Range[0, 107] (* Ant King, Oct 19 2010 *)

PROG

(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

CROSSREFS

Cf. A000290, A124978, A006431, A180149, A245022.

Sequence in context: A033182 A053797 A254011 * A275806 A228369 A108244

Adjacent sequences:  A002632 A002633 A002634 * A002636 A002637 A002638

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified March 29 06:34 EDT 2017. Contains 284250 sequences.