A268241 Number of closed factors of length n in Thue-Morse sequence A010060.

1, 2, 2, 2, 4, 4, 6, 4, 8, 8, 10, 8, 12, 8, 8, 8, 16, 16, 16, 16, 18, 16, 20, 24, 20, 16, 16, 16, 16, 16, 24, 32, 32, 32, 32, 32, 32, 32, 34, 36, 34, 32, 36, 40, 44, 48, 44, 40, 36, 32, 32, 32, 32, 32, 32, 32, 32, 32, 40, 48, 56, 64, 64, 64, 64, 64, 64, 64, 64

Offset

1,2

Links

Lars Blomberg, Table of n, a(n) for n = 1..10000

Luke Schaeffer, Jeffrey Shallit, Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences, Electronic Journal of Combinatorics 23(1) (2016), #P1.25.

Formula

Th. 2 of Schaeffer-Shallit gives explicit formula.

For n >= 8 and integer k >= -1

a) if (15*2^k < n <= 18*2^k) then a(n) = 2^(k+4);

b) if (18*2^k < n <= 19*2^k) then a(n) = 2*n - 20*2^k - 2;

c) if (19*2^k < n <= 20*2^k) then a(n) = 56*2^k - 2*n + 2;

d) if (20*2^k < n <= 22*2^k) then a(n) = 4*n - 64*2^k - 4;

e) if (22*2^k < n <= 24*2^k) then a(n) = 112*2^k - 4*n + 4;

f) if (24*2^k < n <= 28*2^k) then a(n) = 2^(k+4);

g) if (28*2^k < n <= 30*2^k) then a(n) = 8*n - 208*2^k - 8;

Example

For n=40 we must select k=1 and equation c) so a(40) = 56*2^k - 2*n + 2 = 56*2^1 - 2*40 + 2 = 34.
For n=41 we must select k=1 and equation d) so a(41) = 4*n - 64*2^k - 4 = 4*41 - 64*2^1 - 4 = 32.

Crossrefs

Cf. A010060.

Sequence in context: A274143 A166271 A237520 * A134318 A246452 A104295

Adjacent sequences: A268238 A268239 A268240 * A268242 A268243 A268244

Keyword

nonn

Author

N. J. A. Sloane, Feb 06 2016

Extensions

More terms from Lars Blomberg, Feb 09 2016

Status

approved