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

0,2

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

Links

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

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.

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A341982 A337686 A007978 * A333600 A339625 A096737

Adjacent sequences: A245572 A245573 A245574 * A245576 A245577 A245578

Keyword

nonn

Author

Reinhard Zumkeller, Aug 04 2014

Status

approved