A245575 Number of ways of writing n as the sum of two quarter-squares (cf. A002620).

1, 2, 3, 2, 3, 2, 4, 2, 3, 2, 4, 2, 3, 4, 2, 2, 4, 2, 5, 0, 4, 4, 4, 0, 3, 4, 4, 2, 2, 4, 2, 4, 5, 0, 4, 0, 6, 4, 2, 2, 3, 2, 6, 2, 2, 4, 4, 0, 4, 2, 5, 4, 2, 2, 2, 4, 4, 2, 6, 0, 3, 4, 4, 0, 2, 6, 4, 2, 4, 2, 2, 0, 7, 4, 4, 0, 6, 0, 4, 2, 2, 6, 2, 2, 5, 4

Offset: 0

Author

Reinhard Zumkeller, Aug 04 2014

Keywords

• nonn

Comments

a(n) is also the number of times n appears in the triangle A338796, or equivalently, the number of positive integer solutions of the equation A338796(x, y) = n for y <= x. - Stefano Spezia, Mar 03 2022

Examples

			a(10) = #{9+1, 6+4, 4+6, 1+9} = 4;
a(11) = #{9+2, 2+9} = 2;
a(12) = #{12+0, 6+6, 0+12} = 3;
a(13) = #{12+1, 9+4, 4+9, 1+12} = 4;
a(14) = #{6+1, 1+6} = 2;
a(15) = #{9+6, 6+9} = 2;
a(16) = #{16+0, 12+4, 4+12, 0+16} = 4;
a(17) = #{16+1, 1+16} = 2;
a(18) = #{16+2, 12+6, 9+9, 6+12, 2+16} = 5;
a(19) = #{} = 0;
a(20) = #{20+0, 16+4, 4+16, 0+20} = 4.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A002620, A007550, A182834, A240025, A240952, A245585, A338796.

Programs

Haskell

a245575 n = a245575_list !! n
a245575_list = f 0 [] $ tail a002620_list where
   f u vs ws'@(w:ws)
     | u < w     = (sum $ map (a240025 . (u -)) vs) : f (u + 1) vs ws'
     | otherwise = f u (w : vs) ws

Mathematica

qsQ[n_] := qsQ[n] = With[{s = Sqrt[n]}, Which[IntegerQ[s], True, n == Floor[s] (Floor[s]+1), True, True, False]]; a[n_] := Count[Range[0, n], k_ /; qsQ[k] && qsQ[n-k]]; Array[a, 100, 0] (* Jean-François Alcover, May 08 2017 *) (* or *)
u[{x_,y_}] := 2-Boole[x==y]; a[n_] := Total[u /@ IntegerPartitions[n, {2}, Floor[Range[1 + 2 Sqrt@ n]^2/4]]]; Array[a, 100, 0] (* Giovanni Resta, May 08 2017 *)

Formula

a(A182834(n)) mod 2 = 0; a(A007550(n)) mod 2 = 1;

a(A240952(n)) = n and a(A240952(m)) <> n for m < a(n);

a(A245585(n)) = 0.