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A025427 Number of partitions of n into 3 nonzero squares. 37
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,28
COMMENTS
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
LINKS
R. J. Mathar and R. Zumkeller, Table of n, a(n) for n = 0..10000, first 5592 terms from R. J. Mathar
FORMULA
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
EXAMPLE
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).
MAPLE
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
MATHEMATICA
Count[PowersRepresentations[#, 3, 2], pr_ /; (Times @@ pr) > 0]& /@ Range[0, 120] (* Jean-François Alcover, Jan 30 2018 *)
PROG
(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
CROSSREFS
Cf. A000408, A024795, A223730 (multiplicities for the primitive case). - Wolfdieter Lang, Apr 03 2013
Column k=3 of A243148.
Sequence in context: A343653 A066620 A219023 * A348536 A245963 A291375
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)