A394938 a(n) = floor(log_2(2*n - 1)) + floor(log_2(2*n - 1 - 2^(floor(log_2(2*n - 2)) - 1))) + 2.

4, 5, 6, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16

Offset

2,1

Comments

For n >= 3, a(n) is the tree-depth of the n-antiprism graph.

The n-antiprism graph is isomorphic to the square of the cycle C_{2n}.

Links

Table of n, a(n) for n=2..97.

Eric Weisstein's World of Mathematics, Antiprism Graph.

Eric Weisstein's World of Mathematics, Tree-Depth.

Formula

a(n) = floor(log_2(2*n - 1)) + floor(log_2(2*n - 1 - 2^(floor(log_2(2*n - 2)) - 1))) + 2 for n >= 2.

Example

a(2) = 4 since floor(log_2(3)) + floor(log_2(2)) + 2 = 4.

Mathematica

Table[Floor[Log2[2 n - 1]] + Floor[Log2[2 n - 1 - 2^(Floor[Log2[2 n - 2]] - 1)]] + 2, {n, 2, 20}]

Crossrefs

Cf. A395540 (tree-depths of prism graphs).

Sequence in context: A395328 A067471 A102691 * A014553 A394856 A227422

Adjacent sequences: A394934 A394935 A394936 * A394939 A394940 A394941

Keyword

nonn,easy,new

Author

Eric W. Weisstein, Jul 01 2026

Status

approved