A025426 Number of partitions of n into 2 nonzero squares.

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

Offset

0,51

Comments

For records see A007511, A048610, A016032. - R. J. Mathar, Feb 26 2008

Links

Robin Jones, Table of n, a(n) for n = 0..20000 (Terms 0..10000 from Reinhard Zumkeller).

Vaclav Kotesovec, Graph - the asymptotic ratio (10^8 terms)

Index entries for sequences related to sums of squares

Formula

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(A018825(n)) = 0; a(A000404(n)) > 0; a(A025284(n)) = 1; a(A007692(n)) > 1. - Reinhard Zumkeller, Aug 16 2011

a(A000578(n)) = A084888(n). - Reinhard Zumkeller, Jul 18 2012

a(n) = Sum_{i=1..floor(n/2)} A010052(i) * A010052(n-i). - Wesley Ivan Hurt, Apr 19 2019

a(n) = [x^n y^2] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019

Conjecture: Sum_{k=1..n} a(k) ~ n*Pi/8. - Vaclav Kotesovec, Dec 28 2023

Maple

A025426 := proc(n)
    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:
end proc: # R. J. Mathar, Sep 15 2015

Mathematica

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 *)

Prog

(Haskell)
a025426 n = sum $ map (a010052 . (n -)) $
                      takeWhile (<= n `div` 2) $ tail a000290_list
a025426_list = map a025426 [0..]
-- Reinhard Zumkeller, Aug 16 2011
(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

Crossrefs

Cf. A000161 (2 nonnegative squares), A063725 (order matters), A025427 (3 nonzero squares).

Cf. A172151, A004526. - Reinhard Zumkeller, Jan 26 2010

Column k=2 of A243148.

Sequence in context: A285313 A231366 A158924 * A269244 A204246 A053200

Adjacent sequences: A025423 A025424 A025425 * A025427 A025428 A025429

Keyword

nonn,easy

Author

David W. Wilson

Status

approved