This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A002635 Number of partitions of n into 4 squares. (Formerly M0053 N0018) 24

%I M0053 N0018

%S 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,

%T 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,

%U 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

%N Number of partitions of n into 4 squares.

%C a(A124978(n)) = n; a(A006431(n)) = 1; a(A180149(n)) = 2; a(A245022(n)) = 3. - _Reinhard Zumkeller_, Jul 13 2014

%D 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.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002635/b002635.txt">Table of n, a(n) for n = 0..10000</a>

%H E. Grosswald, <a href="http://dx.doi.org/10.1007/978-1-4613-8566-0_8">The Problem of the Uniqueness of Essentially Distinct Representations</a>, in Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 84.

%H D. H. Lehmer, <a href="http://dx.doi.org/10.1090/S0025-5718-47-99581-1">Review of Loria article</a>, Math. Comp. 2 (1947), 301-302.

%H Gino Loria, <a href="/A002635/a002635.pdf"> Sulla scomposizione di un intero nella somma di numeri poligonali. (Italian).</a> 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]

%H M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00159-4">Some formulas for partitions into squares</a>, Discrete Math, 211 (2000), pp. 225-228.

%H James A. Sellers, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Sellers/sellers58.html">Partitions Excluding Specific Polygonal Numbers As Parts</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%e 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.

%t Length[PowersRepresentations[ #, 4, 2]] & /@ Range[0, 107] (* _Ant King_, Oct 19 2010 *)

%o (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))))),","))

%o (PARI) a(n)=local(c=0); if(n>=0, forvec(x=vector(4,k,[0,sqrtint(n)]),c+=norml2(x)==n,1)); c

%o a002635 = p (tail a000290_list) 4 where

%o p ks'@(k:ks) c m = if m == 0 then 1 else

%o if c == 0 || m < k then 0 else p ks' (c - 1) (m - k) + p ks c m

%o -- _Reinhard Zumkeller_, Jul 13 2014

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

%Y 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).

%K nonn,easy,nice

%O 0,5

%A _N. J. A. Sloane_.

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 9 17:18 EST 2019. Contains 329879 sequences. (Running on oeis4.)