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A025427
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Number of partitions of n into 3 nonzero squares.
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37
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0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 2, 0, 1, 1, 0, 0, 2, 1, 1, 1, 0, 2, 0, 0, 2, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 0, 1, 3, 0, 1, 2, 0, 2, 0, 1, 2, 0, 0, 1, 3, 1, 1, 2, 1, 0, 1, 1, 2, 2, 1, 2, 1, 0, 0, 3, 1, 2, 1, 0, 3, 0, 1, 3, 2, 1, 0, 1, 2, 0, 1, 1, 2, 3, 0, 3, 2, 0, 1, 2, 1, 2
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OFFSET
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0,28
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COMMENTS
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The non-vanishing values a(n) give the multiplicities for the numbers n appearing in A000408. See also A024795 where these numbers n are listed a(n) times. For the primitive case see A223730 and A223731. - Wolfdieter Lang, Apr 03 2013
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LINKS
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R. J. Mathar and R. Zumkeller, Table of n, a(n) for n = 0..10000, first 5592 terms from R. J. Mathar
Index to sequences related to sums of squares and cubes.
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FORMULA
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a(A004214(n) = 0; a(A000408(n) > 0; a(A025414(n)) = n and a(m) != n for m < A025414(n). - Reinhard Zumkeller, Feb 26 2015
a(4n) = a(n). This is because if a number divisible by 4 is the sum of three squares, each of those squares must be even. - Robert Israel, Mar 09 2016
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} A010052(i) * A010052(k) * A010052(n-i-k). - Wesley Ivan Hurt, Apr 19 2019
a(n) = [x^n y^3] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019
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EXAMPLE
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a(27) = 2 because 1^2 + 1^2 + 5^2 = 27 = 3^2 + 3^2 + 3^2. The second representation is not primitive (gcd(3,3,3) = 3 not 1).
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MAPLE
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A025427 := proc(n)
local a, x, y, zsq ;
a := 0 ;
for x from 1 do
if 3*x^2 > n then
return a;
end if;
for y from x do
if x+2*y^2 > n then
break;
end if;
zsq := n-x^2-y^2 ;
if issqr(zsq) then
a := a+1 ;
end if;
end do:
end do:
end proc: # R. J. Mathar, Sep 15 2015
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MATHEMATICA
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Count[PowersRepresentations[#, 3, 2], pr_ /; (Times @@ pr) > 0]& /@ Range[0, 120] (* Jean-François Alcover, Jan 30 2018 *)
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PROG
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(Haskell)
a025427 n = sum $ map f zs where
f x = sum $ map (a010052 . (n - x -)) $
takeWhile (<= div (n - x) 2) $ dropWhile (< x) zs
zs = takeWhile (< n) $ tail a000290_list
-- Reinhard Zumkeller, Feb 26 2015
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CROSSREFS
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Cf. A000408, A024795, A223730 (multiplicities for the primitive case). - Wolfdieter Lang, Apr 03 2013
Column k=3 of A243148.
Cf. A000290, A010052, A004214, A025321, A025414, A025426.
Sequence in context: A343653 A066620 A219023 * A348536 A245963 A291375
Adjacent sequences: A025424 A025425 A025426 * A025428 A025429 A025430
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KEYWORD
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nonn,easy
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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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