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A025428
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Number of partitions of n into 4 nonzero squares.
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40
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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
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OFFSET
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0,29
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COMMENTS
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Records occur at n= 4, 28, 52, 82, 90, 130, 162, 198, 202, 210,.... - R. J. Mathar, Sep 15 2015
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LINKS
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FORMULA
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a(n) = [x^n y^4] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019
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MAPLE
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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:
# 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):
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MATHEMATICA
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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 *)
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PROG
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(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))))
(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 )))
(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), ", "))
(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<j&&i-k*d*d>0, a[i-k*d*d, j-k], if(k==j&&i-k*d*d==0, 1))))); a[d*d, 1]=1); for(i=1, n, print(i" "a[i, 4]))} /* Robert Gerbicz, Sep 28 2012 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Values of a(0..10^4) double-checked by M. F. Hasler, Sep 17 2012
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STATUS
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approved
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