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A363710 a(n) is the number of pairs of nonnegative integers (x, y) such that x + y = n and A003188(x) AND A003188(y) = 0 (where AND denotes the bitwise AND operator).

%I A363710 #11 Jun 02 2025 16:49:53
%S A363710 1,2,2,2,4,4,2,2,4,8,6,4,6,4,2,2,4,8,10,8,12,12,6,4,6,12,8,4,6,4,2,2,
%T A363710 4,8,10,8,16,20,10,8,12,24,20,12,16,12,6,4,6,12,16,12,16,16,8,4,6,12,
%U A363710 8,4,6,4,2,2,4,8,10,8,16,20,10,8,16,32,28,20,28
%N A363710 a(n) is the number of pairs of nonnegative integers (x, y) such that x + y = n and A003188(x) AND A003188(y) = 0 (where AND denotes the bitwise AND operator).
%C A363710 Equivalently, a(n) is the number of k >= 0 such that A332497(k) + A332498(k) = n.
%C A363710 The set of pairs of nonnegative integers (x, y) such that A003188(x) AND A003188(y) = 0 is related to the T-square fractal (see illustration in Links section).
%H A363710 Rémy Sigrist, <a href="/A363710/b363710.txt">Table of n, a(n) for n = 0..8192</a>
%H A363710 Rémy Sigrist, <a href="/A332497/a332497.png">Scatterplot of (x, y) such that x, y < 2^10 and A003188(x) AND A003188(y) = 0</a>
%H A363710 Wikipedia, <a href="https://en.wikipedia.org/wiki/T-square_(fractal)">T-square (fractal)</a>
%F A363710 a(n) = 2 iff n belongs to A075427.
%e A363710 For n = 8:
%e A363710 - we have:
%e A363710   k  A332497(8-k)  A332497(k)  A332497(8-k) AND A332497(k)
%e A363710   -  ------------  ----------  ---------------------------
%e A363710   0            12           0                            0
%e A363710   1             4           1                            0
%e A363710   2             5           3                            1
%e A363710   3             7           2                            2
%e A363710   4             6           6                            6
%e A363710   5             2           7                            2
%e A363710   6             3           5                            1
%e A363710   7             1           4                            0
%e A363710   8             0          12                            0
%e A363710 - so a(8) = 4.
%o A363710 (PARI) a(n) = 2*sum(k=0, n\2, bitand(bitxor(n-k, (n-k)\2), bitxor(k, k\2))==0) - (n==0)
%o A363710 (Python) A363710=lambda n: sum(map(lambda k: not (k^k>>1)&(n-k^n-k>>1),range(n+1>>1)))<<1 if n else 1 # _Natalia L. Skirrow_, Jun 22 2023
%Y A363710 Cf. A003188, A075427, A332497, A332498.
%K A363710 nonn,base
%O A363710 0,2
%A A363710 _Rémy Sigrist_, Jun 17 2023