A087917 Number of unordered ways to write n as a sum of 3 odious numbers (A000069).

0, 0, 1, 1, 1, 2, 1, 1, 2, 3, 2, 3, 3, 2, 4, 6, 5, 5, 6, 5, 6, 8, 8, 9, 9, 8, 10, 12, 12, 14, 14, 10, 14, 19, 14, 18, 20, 14, 19, 25, 21, 20, 27, 22, 23, 32, 26, 27, 31, 29, 31, 36, 35, 35, 39, 34, 38, 47, 40, 42, 47, 40, 43, 60, 53, 44, 60, 50, 48, 68, 62, 54, 64, 65, 58, 75

Offset: 1

Author

Ralf Stephan, Oct 18 2003

Keywords

• nonn
• easy

Examples

			a(6) = 2 as 6 = 1 + 1 + 4 = 2 + 2 + 2. 1, 2 and 4 are odious as the number of ones in the binary expansion is odd. The partition 1 + 2 + 3 does not count as 3 is not odious; the number of ones in the binary expansion of 3 is 2 (even). - _David A. Corneth_, Apr 23 2025
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 200 terms from Robert Price)

Crossrefs

Cf. A000069, A087916.

Cf. A010060.

Programs

Mathematica

A010060 = Cases[Import["https://oeis.org/A010060/b010060.txt", "Table"], {, }][[All, 2]];
Table[Length@Select[DeleteDuplicates[Sort /@ Select[Tuples[Range[n], 3], Total[#] == n &]], A010060[[#[[1]] + 1]]*A010060[[#[[2]] + 1]]* A010060[[#[[3]] + 1]] == 1 &], {n, 200}] (* Robert Price, Apr 22 2025 *)

PARI

a(n)=sum(i=1,n,sum(j=1,i,sum(k=1,j,if(i+j+k-n,0, A010060(i)*A010060(j)*A010060(k)))))

PARI

first(n) = {
    res = vector(n);
    for(i = 1, n\3,
        if(bitand(hammingweight(i), 1),
            for(j = i, (n - i)\2,
                if(bitand(hammingweight(j),1),
                    for(k = j, n - i - j,
                        res[i+j+k]+=bitand(hammingweight(k),1)))))); res
} \\ David A. Corneth, Apr 23 2025

Formula

a(n) = Sum_{i+j+k = n} A010060(i)*A010060(j)*A010060(k). - Benoit Cloitre, Mar 19 2004