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A025441 Number of partitions of n into 2 distinct nonzero squares.

%I A025441 #65 Feb 18 2024 02:04:04
%S A025441 0,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,1,0,0,1,0,0,0,0,1,1,0,0,1,0,0,0,0,
%T A025441 1,0,0,1,0,0,1,1,0,0,0,1,0,0,0,0,1,0,1,1,0,0,0,0,1,0,0,1,0,0,0,2,0,0,
%U A025441 1,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,0,2,0,0,0,1,1,0,0,0,0,0,0,1,0,0,1,1,0,0,1,0,1,0
%N A025441 Number of partitions of n into 2 distinct nonzero squares.
%H A025441 T. D. Noe, <a href="/A025441/b025441.txt">Table of n, a(n) for n = 0..10000</a>
%H A025441 Michael Gilleland, <a href="/selfsimilar.html">Some Self-Similar Integer Sequences</a>
%H A025441 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A025441 a(A025302(n)) = 1. - _Reinhard Zumkeller_, Dec 20 2013
%F A025441 a(n) = Sum_{ m: m^2|n } A157228(n/m^2). - _Andrey Zabolotskiy_, May 07 2018
%F A025441 a(n) = [x^n y^2] Product_{k>=1} (1 + y*x^(k^2)). - _Ilya Gutkovskiy_, Apr 22 2019
%F A025441 a(n) = Sum_{i=1..floor((n-1)/2)} c(i) * c(n-i), where c is the square characteristic (A010052). - _Wesley Ivan Hurt_, Nov 26 2020
%F A025441 a(n) = A000161(n) - A093709(n). - _Andrey Zabolotskiy_, Apr 12 2022
%t A025441 Table[Count[PowersRepresentations[n, 2, 2], pr_ /; Unequal @@ pr && FreeQ[pr, 0]], {n, 0, 107}] (* _Jean-François Alcover_, Mar 01 2019 *)
%o A025441 (Haskell)
%o A025441 a025441 n = sum $ map (a010052 . (n -)) $
%o A025441                       takeWhile (< n `div` 2) $ tail a000290_list
%o A025441 -- _Reinhard Zumkeller_, Dec 20 2013
%o A025441 (PARI) a(n)=if(n>4,sum(k=1,sqrtint((n-1)\2),issquare(n-k^2)),0) \\ _Charles R Greathouse IV_, Jun 10 2016
%o A025441 (PARI) a(n)=if(n<5,return(0)); 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-issquare(n/2) \\ _Charles R Greathouse IV_, Jun 10 2016
%o A025441 (Python)
%o A025441 from math import prod
%o A025441 from sympy import factorint
%o A025441 def A025441(n):
%o A025441     f = factorint(n).items()
%o A025441     return -int(not (any((e-1 if p == 2 else e)&1 for p,e in f) or n&1)) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1) if n else 0 # _Chai Wah Wu_, Sep 08 2022
%Y A025441 Cf. A060306 gives records; A052199 gives where records occur.
%Y A025441 Cf. A000161, A000290, A010052, A025435, A157228, A053866, A145393, A093709.
%Y A025441 Column k=2 of A341040.
%Y A025441 Cf. A004439 (a(n)=0), A025302 (a(n)=1), A025303 (a(n)=2), A025304 (a(n)=3), A025305 (a(n)=4), A025306 (a(n)=5), A025307 (a(n)=6), A025308 (a(n)=7), A025309 (a(n)=8), A025310 (a(n)=9), A025311 (a(n)=10), A004431 (a(n)>0).
%K A025441 nonn
%O A025441 0,66
%A A025441 _David W. Wilson_