cp's OEIS Frontend

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

%I A245575 #22 Mar 13 2022 18:55:59
%S A245575 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,
%T A245575 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,
%U A245575 4,2,2,0,7,4,4,0,6,0,4,2,2,6,2,2,5,4
%N A245575 Number of ways of writing n as the sum of two quarter-squares (cf. A002620).
%C A245575 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
%H A245575 Reinhard Zumkeller, <a href="/A245575/b245575.txt">Table of n, a(n) for n = 0..10000</a>
%F A245575 a(A182834(n)) mod 2 = 0; a(A007550(n)) mod 2 = 1;
%F A245575 a(A240952(n)) = n and a(A240952(m)) <> n for m < a(n);
%F A245575 a(A245585(n)) = 0.
%e A245575 a(10) = #{9+1, 6+4, 4+6, 1+9} = 4;
%e A245575 a(11) = #{9+2, 2+9} = 2;
%e A245575 a(12) = #{12+0, 6+6, 0+12} = 3;
%e A245575 a(13) = #{12+1, 9+4, 4+9, 1+12} = 4;
%e A245575 a(14) = #{6+1, 1+6} = 2;
%e A245575 a(15) = #{9+6, 6+9} = 2;
%e A245575 a(16) = #{16+0, 12+4, 4+12, 0+16} = 4;
%e A245575 a(17) = #{16+1, 1+16} = 2;
%e A245575 a(18) = #{16+2, 12+6, 9+9, 6+12, 2+16} = 5;
%e A245575 a(19) = #{} = 0;
%e A245575 a(20) = #{20+0, 16+4, 4+16, 0+20} = 4.
%t A245575 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 *)
%t A245575 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 *)
%o A245575 (Haskell)
%o A245575 a245575 n = a245575_list !! n
%o A245575 a245575_list = f 0 [] $ tail a002620_list where
%o A245575    f u vs ws'@(w:ws)
%o A245575      | u < w     = (sum $ map (a240025 . (u -)) vs) : f (u + 1) vs ws'
%o A245575      | otherwise = f u (w : vs) ws
%Y A245575 Cf. A002620, A007550, A182834, A240025, A240952, A245585, A338796.
%K A245575 nonn
%O A245575 0,2
%A A245575 _Reinhard Zumkeller_, Aug 04 2014