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A106634
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Number of ways to express n as i*j+k*l, with i,j,k,l in the range [0..n].
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5
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1, 6, 21, 32, 62, 58, 124, 88, 173, 158, 226, 156, 380, 194, 340, 356, 466, 274, 613, 316, 690, 536, 596, 404, 1060, 552, 734, 728, 1032, 546, 1376, 596, 1213, 932, 1026, 976, 1858, 750, 1180, 1144, 1910, 854, 2048, 908, 1784, 1730, 1500, 1016, 2800
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OFFSET
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0,2
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COMMENTS
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Number of ordered 4-tuples [i,j,k,l] with n=i*j+k*l and i,j,k,l in the range [0..n].
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LINKS
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FORMULA
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a(n) = (4*n + 2)*tau(n) + Sum_{i=1..n-1} tau(i)*tau(n-i), for n>0 ;
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EXAMPLE
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a(1)=6: the 4-tuples ijkl are 1100, 1101, 1110, 0011, 0111, 1011.
a(2)=21: 1111, 2100, 210x, 21x0, 1200, 120x, 12x0, where x = 1 or 2, and ten more with the two halves swapped.
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MAPLE
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local a, i, j, k, l ;
a := 0 ;
for i from 0 to n do
for j from 0 to n do
if i*j > n then
break;
end if;
for k from 0 to n do
if i*j = n then
# treat l=0 separately
a := a+1 ;
end if;
# l=1..n
if k =0 then
if i*j=n then
a := a+n ;
end if;
else
l := (n-i*j)/k ;
if l >=1 and l <=n and type(l, 'integer') then
a := a+1 ;
end if;
end if;
end do:
end do:
end do:
a ;
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MATHEMATICA
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list[n_] := Module[{v, i, j}, v[_] = 0;
For[i = 2, i <= n, i++, For[j = 1, j <= Min[Quotient[n, i], i-1], j++, v[i*j]+= 2]];
For[i = 1, i <= Floor@Sqrt[n], i++, v[i^2]++];
Join[{1}, Table[2 Sum[v[j] v[i-j], {j, Quotient[i, 2]+1, i-1}]+If[OddQ[i], 0, v[i/2]^2]+(4i+2) v[i], {i, 1, n}]]];
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PROG
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(PARI) list(n)={
my(v=vector(n));
for(i=2, n, for(j=1, min(n\i, i-1), v[i*j]+=2));
for(i=1, sqrtint(n), v[i^2]++);
concat(1, vector(n, i, 2*sum(j=i\2+1, i-1, v[j]*v[i-j])+if(i%2, , v[i/2]^2)+(4*i+2)*v[i]))
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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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STATUS
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approved
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