A025428 Number of partitions of n into 4 nonzero squares.
0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 3, 0, 1, 2, 0, 1, 2, 1, 2, 2, 1, 2, 1, 0, 3, 2, 1, 2, 1, 2, 1, 2, 2, 1, 4, 1, 2, 3, 0, 2, 4, 1, 3, 2, 1, 4, 1, 1, 3, 3, 2, 2, 4, 2, 1, 3, 2, 3, 4, 2, 3, 3, 1, 2, 5, 2, 4, 3, 2, 4, 1, 1, 6, 4, 3, 4, 2, 3, 0, 4, 4, 3, 5, 1, 5, 5, 1, 4, 5, 2
Offset: 0
Links
Crossrefs
Programs
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Maple
A025428 := proc(n) local a,i,j,k,lsq ; a := 0 ; for i from 1 do if 4*i^2 > n then return a; end if; for j from i do if i^2+3*j^2 > n then break; end if; for k from j do if i^2+j^2+2*k^2 > n then break; end if; lsq := n-i^2-j^2-k^2 ; if lsq >= k^2 and issqr(lsq) then a := a+1 ; end if; end do: end do: end do: end proc: seq(A025428(n),n=1..40) ; # R. J. Mathar, Jun 15 2018 # second Maple program: b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+`if`(i^2>n, 0, b(n-i^2, i, t-1)))) end: a:= n-> b(n, isqrt(n), 4): seq(a(n), n=0..100); # Alois P. Heinz, Apr 14 2019
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Mathematica
nn = 100; lim = Sqrt[nn]; t = Table[0, {nn}]; Do[n = a^2 + b^2 + c^2 + d^2; If[n <= nn, t[[n]]++], {a, lim}, {b, a, lim}, {c, b, lim}, {d, c, lim}]; t (* T. D. Noe, Sep 28 2012 *) f[n_] := Length@ IntegerPartitions[n, {4}, Range[ Floor[ Sqrt[n - 1]]]^2]; Array[f, 105] (* Robert G. Wilson v, Sep 28 2012 *) -
PARI
A025428(n)=sum(a=1,n,sum(b=1,a,sum(c=1,b,sum(d=1,c,a^2+b^2+c^2+d^2==n))))
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PARI
A025428(n)=sum(a=1,sqrtint(max(n-3,0)), sum(b=1,min(sqrtint(n-a^2-2),a), sum(c=1,min(sqrtint(n-a^2-b^2-1),b),issquare(n-a^2-b^2-c^2,&d) & d <= c )))
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PARI
A025428(n)=sum(a=sqrtint(max(n,4)\4),sqrtint(max(n-3,0)), sum(b=sqrtint((n-a^2)\3-1)+1,min(sqrtint(n-a^2-2),a), sum(c=sqrtint((t=n-a^2-b^2)\2-1)+1, min(sqrtint(t-1),b), issquare(t-c^2) ))) \\ - M. F. Hasler, Sep 17 2012 for(n=1,100,print1(A025428(n),","))
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PARI
T(n)={a=matrix(n,4,i,j,0);for(d=1,sqrtint(n),forstep(i=n,d*d+1,-1,for(j=2,4,a[i,j]+=sum(k=1,j,if(k
Formula
For n>0, a(n) = ( A063730(n) + 6*A213024(n) + 3*A063725(n/2) + 8*A092573(n) + 6*A010052(n/4) ) / 24. - Max Alekseyev, Sep 30 2012
a(n) = ( A000118(n) - 4*A005875(n) - 6*A004018(n) - 12*A000122(n) - 15*A000007(n) + 12*A014455(n) - 24*A033715(n) - 12*A000122(n/2) + 12*A004018(n/2) + 32*A033716(n) - 32*A000122(n/3) + 48*A000122(n/4) ) / 384. - Max Alekseyev, Sep 30 2012
a(n) = [x^n y^4] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} A010052(i) * A010052(j) * A010052(k) * A010052(n-i-j-k). - Wesley Ivan Hurt, Apr 19 2019
Extensions
Values of a(0..10^4) double-checked by M. F. Hasler, Sep 17 2012
Comments