A097384 Total number of comparisons to find each of the values 1 through n using a binary search with 3-way comparisons (less than, equal and greater than), always choosing the mid-most value to compare to.

0, 2, 3, 6, 9, 11, 13, 17, 21, 25, 29, 32, 35, 38, 41, 46, 51, 56, 61, 66, 71, 76, 81, 85, 89, 93, 97, 101, 105, 109, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 203, 209, 214, 219, 224, 229, 234, 239, 244, 249, 254, 259, 264, 269, 274

Offset: 1

Author

Franklin T. Adams-Watters, Aug 11 2004

Keywords

• easy
• nonn

Comments

Pure binary search with equality.

Examples

			a_5 = 9 = 5 + a_2 + a_2; this is the smallest example where choosing the middle value is not optimal.
		

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

See A097383 for the sequence with an optimized binary search. A003314 is the sequence with only 2-way comparisons.

Formula

n + a_{floor((n-1)/2)} + a_{ceiling((n-1)/2)}.