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

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, 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, 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

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• Michael Gilleland, Some Self-Similar Integer Sequences
• Index entries for sequences related to sums of squares

Crossrefs

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

Cf. A000161, A000290, A010052, A025435, A157228, A053866, A145393, A093709.

Column k=2 of A341040.

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).

Programs

Haskell

a025441 n = sum $ map (a010052 . (n -)) $
                      takeWhile (< n `div` 2) $ tail a000290_list
-- Reinhard Zumkeller, Dec 20 2013

Mathematica

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

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

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

Python

from math import prod
from sympy import factorint
def A025441(n):
    f = factorint(n).items()
    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

Formula

a(A025302(n)) = 1. - Reinhard Zumkeller, Dec 20 2013

a(n) = Sum_{ m: m^2|n } A157228(n/m^2). - Andrey Zabolotskiy, May 07 2018

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

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

a(n) = A000161(n) - A093709(n). - Andrey Zabolotskiy, Apr 12 2022