A078881 Size of the largest subset S of {1,2,3,...,n} with the property that if i and j are distinct elements of S then i XOR j is not in S, where XOR is the bitwise exclusive-OR operator.

1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 8, 8, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44

Offset: 1

Author

John W. Layman, Dec 11 2002

Keywords

• nonn

Comments

Is this sequence the same as A006165?

The answer is yes, as shown by Hsien-Kuei Hwang, S Janson, TH Tsai (2016). More precisely, a(n) = A006165(n+1) for all n >= 1. - N. J. A. Sloane, Nov 26 2017

Can be formulated as an integer linear program: maximize sum {i = 1 to n} x[i] subject to x[i] + x[j] + x[i XOR j] <= 2 for all i < j, x[i] in {0,1} for all i. - Rob Pratt, Feb 09 2010

a(n) = A006165(n+1) checked for n <= 1023. - Rob Pratt, Dec 04 2014

References

• Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585

Crossrefs

Cf. A078882.

Same (apart from offset) as A006165.

Formula

a(n) = A006165(n+1) for all n >= 1. - N. J. A. Sloane, Nov 26 2017

Extensions

More terms from Rob Pratt, Feb 09 2010