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A025435
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Number of partitions of n into 2 distinct squares.
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7
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0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0
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OFFSET
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0,26
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COMMENTS
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a(A004435(n)) = 0; a(A001983(n)) > 0. - Reinhard Zumkeller, Dec 20 2013
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
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FORMULA
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a(n) = A000161(n) - A010052(2n). - M. F. Hasler, Aug 05 2018
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EXAMPLE
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G.f. = x + x^4 + x^5 + x^9 + x^10 + x^13 + x^16 + x^17 + x^20 + 2*x^25 + ...
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MAPLE
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A025435 := proc(n)
local i, j, ans;
ans := 0;
for i from 0 to n do
for j from i+1 to n do
if i^2+j^2=n then
ans := ans+1
fi
end do
end do;
ans ;
end proc: # R. J. Mathar, Aug 04 2018
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, Sum[ Boole[ n == i^2 + j^2], {i, Sqrt[n]}, {j, 0, i - 1}]]; (* Michael Somos, Jun 24 2015 *)
a[ n_] := Length@ PowersRepresentations[ n, 2, 2] - Boole @ IntegerQ @ Sqrt[2 n]; (* Michael Somos, Jun 24 2015 *)
a[ n_] := SeriesCoefficient[ With[ {f = (EllipticTheta[ 3, 0, x] + 1)/2, g = (EllipticTheta[ 3, 0, x^2] + 1)/2}, f f - g] / 2, {x, 0, n}]; (* Michael Somos, Jun 24 2015 *)
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PROG
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(Haskell)
a025435 0 = 0
a025435 n = a010052 n + sum
(map (a010052 . (n -)) $ takeWhile (< n `div` 2) $ tail a000290_list)
-- Reinhard Zumkeller, Dec 20 2013
(PARI) {a(n) = if( n<0, 0, sum(i=1, sqrtint(n), sum(j=0, i-1, n == i^2 + j^2)))}; /* Michael Somos, Jun 24 2015 */
(PARI) A025435(n)=sum(k=sqrtint((n-1+!n)\2)+1, sqrtint(n), issquare(n-k^2))-issquare(n/2) \\ or A000161(n)-issquare(n/2). - M. F. Hasler, Aug 05 2018
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CROSSREFS
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Cf. A010052, A000290, A000161, A025441.
Sequence in context: A286562 A185644 A319080 * A304685 A186714 A160382
Adjacent sequences: A025432 A025433 A025434 * A025436 A025437 A025438
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KEYWORD
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nonn
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AUTHOR
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David W. Wilson
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STATUS
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approved
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