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 A025441 Number of partitions of n into 2 distinct nonzero squares. 17

%I

%S 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 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 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 Number of partitions of n into 2 distinct nonzero squares.

%H T. D. Noe, <a href="/A025441/b025441.txt">Table of n, a(n) for n = 0..10000</a>

%H Michael Gilleland, <a href="/selfsimilar.html">Some Self-Similar Integer Sequences</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F a(A025302(n)) = 1. - _Reinhard Zumkeller_, Dec 20 2013

%F a(n) = Sum_{ m: m^2|n } A157228(n/m^2). - _Andrey Zabolotskiy_, May 07 2018

%F a(n) = [x^n y^2] Product_{k>=1} (1 + y*x^(k^2)). - _Ilya Gutkovskiy_, Apr 22 2019

%p P:=proc(n) local a,x; a:=1; x:=0; while a^2<trunc(n/2)

%p do if frac(sqrt(n-a^2))=0 then x:=x+1; fi; a:=a+1; od; x; end:

%p seq(P(i),i=1..100); # _Paolo P. Lava_, Mar 12 2018

%t Table[Count[PowersRepresentations[n, 2, 2], pr_ /; Unequal @@ pr && FreeQ[pr, 0]], {n, 0, 107}] (* _Jean-François Alcover_, Mar 01 2019 *)

%o a025441 n = sum \$ map (a010052 . (n -)) \$

%o takeWhile (< n `div` 2) \$ tail a000290_list

%o -- _Reinhard Zumkeller_, Dec 20 2013

%o (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 (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

%Y Cf. A060306 gives records; A052199 gives where records occur.

%Y Cf. A000161, A000290, A010052, A025435, A157228, A053866, A145393.

%K nonn

%O 0,66

%A _David W. Wilson_

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Last modified December 14 05:36 EST 2019. Contains 329978 sequences. (Running on oeis4.)