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A025428 Number of partitions of n into 4 nonzero squares. 39
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,29

COMMENTS

Records occur at n= 4, 28, 52, 82, 90, 130, 162, 198, 202, 210,.... - R. J. Mathar, Sep 15 2015

LINKS

T. D. Noe, Table of n, a(n) for n = 0..10000

Index entries for sequences related to sums of squares

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

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

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 *)

PROG

(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 */

CROSSREFS

Cf. A000414, A000534, A025357-A025375, A216374, A025416 (greedy inverse).

Column k=4 of A243148.

Sequence in context: A160499 A274876 A065718 * A199176 A021336 A100749

Adjacent sequences:  A025425 A025426 A025427 * A025429 A025430 A025431

KEYWORD

nonn,easy

AUTHOR

David W. Wilson

EXTENSIONS

Values of a(0..10^4) double-checked by M. F. Hasler, Sep 17 2012

STATUS

approved

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Last modified October 21 14:28 EDT 2019. Contains 328301 sequences. (Running on oeis4.)