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A025426
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Number of partitions of n into 2 nonzero squares.
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46
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0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0
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OFFSET
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0,51
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COMMENTS
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LINKS
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FORMULA
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Let m = A004018(n)/4. If m is even then a(n) = m/2, otherwise a(n) = (m - (-1)^A007814(n))/2. - Max Alekseyev, Mar 09 2009, Mar 14 2009
a(n) = [x^n y^2] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019
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MAPLE
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local a, x;
a := 0 ;
for x from 1 do
if 2*x^2 > n then
return a;
end if;
if issqr(n-x^2) then
a := a+1 ;
end if;
end do:
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MATHEMATICA
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m[n_] := m[n] = SquaresR[2, n]/4; a[0] = 0; a[n_] := If[ EvenQ[ m[n] ], m[n]/2, (m[n] - (-1)^IntegerExponent[n, 2])/2]; Table[ a[n], {n, 0, 107}] (* Jean-François Alcover, Jan 31 2012, after Max Alekseyev *)
nmax = 107; sq = Range[Sqrt[nmax]]^2;
Table[Length[Select[IntegerPartitions[n, All, sq], Length[#] == 2 &]], {n, 0, nmax}] (* Robert Price, Aug 17 2020 *)
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PROG
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(Haskell)
a025426 n = sum $ map (a010052 . (n -)) $
takeWhile (<= n `div` 2) $ tail a000290_list
a025426_list = map a025426 [0..]
(PARI) a(n)={my(v=valuation(n, 2), f=factor(n>>v), t=1); for(i=1, #f[, 1], if(f[i, 1]%4==1, t*=f[i, 2]+1, if(f[i, 2]%2, return(0)))); if(t%2, t-(-1)^v, t)/2; } \\ Charles R Greathouse IV, Jan 31 2012
(Python)
from math import prod
from sympy import factorint
def A025426(n): return ((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in factorint(n).items()))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1 # Chai Wah Wu, Jul 07 2022
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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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STATUS
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approved
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