A025425 -id:A025425 - cp's OEIS frontend

A002635 Number of partitions of n into 4 squares.

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

Offset: 0

N. J. A. Sloane

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

			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.

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).

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

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).

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

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

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))))),","))

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

A000174 Number of partitions of n into 5 squares.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 1, 3, 3, 3, 3, 4, 3, 3, 2, 2, 4, 4, 4, 4, 5, 3, 3, 4, 3, 5, 5, 5, 6, 5, 3, 5, 5, 5, 6, 5, 6, 5, 4, 4, 6, 7, 6, 8, 8, 7, 5, 6, 5, 8, 8, 4, 9, 7, 6, 7, 7, 8, 9, 9, 7, 9, 6, 7, 8, 10, 9, 9, 11, 7, 8, 8, 8, 11, 11, 9, 12, 11, 7, 9, 9, 12, 13, 9, 10, 11, 8, 6, 10, 13, 13, 13

Offset: 0

N. J. A. Sloane

nonn

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 84.

Equivalent sequences for other numbers of squares: A010052 (1), A000161 (2), A000164 (3), A002635 (4), A000177 (6), A025422 (7), A025423 (8), A025424 (9), A025425 (10).

Cf. A025429, A295160 (largest number k with a(k) = n).

Table[PowersRepresentations[n, 5, 2] // Length, {n, 0, 100}] (* Jean-François Alcover, Feb 04 2016 *)

A025454 Number of partitions of n into 10 nonnegative cubes.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 1, 2, 2, 2, 3, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 3, 3, 2, 3, 2, 2, 3, 2, 4, 3, 3, 3, 3, 3, 3, 3, 4, 4, 2, 3, 3, 2, 3, 2, 4, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 4, 3, 3, 4, 3, 5, 3, 3, 3, 2, 3, 3, 3, 3

Offset: 0

David W. Wilson

			There are 2 ways to write 34 as a sum of 10 nonnegative cubes: 34 = 4 * 0^3 + 2 * 1^3 + 4 * 2^3 = 2 * 0^2 + 7 * 1^3 + 1 * 2^3. - _David A. Corneth_, Sep 23 2019

David A. Corneth, Table of n, a(n) for n = 0..9999
David A. Corneth, List of the ways to write n as a sum of 10 nonnegative cubes for n = 1..838

Cf. A000144, A025425.

A323891 a(n) is the number of partitions of 72*n + 42 into 10 odd squares.

Original entry on oeis.org

2, 9, 22, 41, 68, 106, 154, 212, 285, 368, 477, 598, 741, 898, 1076, 1286, 1524, 1785, 2068, 2379, 2741, 3131, 3554, 4002, 4497, 5044, 5644, 6274, 6939, 7653, 8445, 9295, 10186, 11117, 12113, 13192, 14355, 15556, 16807, 18147, 19570, 21089, 22673, 24300, 26029, 27865, 29821, 31822, 33894, 36088

Offset: 0

Marius A. Burtea, Feb 12 2019

nonn

			For n=0, 72*0+42 = 42 = 25+9+1+1+1+1+1+1+1+1 = 9+9+9+9+1+1+1+1+1+1, so a(0)=2.
For n=1, 72*1+42 = 114 = 81+25+1+1+1+1+1+1+1+1 = 81+9+9+9+1+1+1+1+1+1 = 49+49+9+1+1+1+1+1+1+1 = 49+25+25+9+1+1+1+1+1+1 = 49+25+9+9+9+9+1+1+1+1 = 49+9+9+9+9+9+9+9+1+1 = 25+25+25+25+9+1+1+1+1+1 = 25+25+25+9+9+9+9+1+1+1 = 25+25+9+9+9+9+9+9+9+1, so a(1)=9.

Laurențiu Panaitopol, Alexandru Gica, Arithmetic problems and number theory, Ed. Gil, Zalău, (2006), ch. 14, p. 85, pr. 32. (in Romanian).

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A000041, A001156, A016754, A025425, A025434, A033461, A035294, A078406, A090677, A167661, A167700.

[#RestrictedPartitions(72*n+42, 10, {(2*d+1)^2:d in [0..100]}): n in [0..100]];

S:= proc(n, k, m)
   option remember;
   local p,j;
   if k = 0 then if n = 0 then return 1 else return 0 fi
   elif m < 1 then return 0
   elif n < k then return 0
   elif n > k*m^2 then return 0
   fi;
   if m^2 > n then
     p:= floor(sqrt(n));
     if p::even then p:= p-1 fi;
     return procname(n, k, p)
   fi;
   add(procname(n-j*m^2,k-j,m-2), j=0..n/m^2)
end proc:
seq(S(72*n+42, 10, 72*n+42), n=0..100); # Robert Israel, Feb 24 2019

a[n_] := IntegerPartitions[72n+42, {10}, Select[ Range[1, 72n+42, 2], IntegerQ@Sqrt@#&]] // Length;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 19 2022 *)

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A002635 Number of partitions of n into 4 squares.

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A000174 Number of partitions of n into 5 squares.

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A025454 Number of partitions of n into 10 nonnegative cubes.

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A323891 a(n) is the number of partitions of 72*n + 42 into 10 odd squares.

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