A293495 Number of balanced binary words of length n whose index is less than (5 + sqrt(5))/2.

1, 2, 4, 8, 12, 18, 24, 34, 44, 54, 66, 76, 86, 98, 110, 116, 126, 140, 158, 166, 172, 184, 196, 208, 222, 238, 226, 228, 230, 228, 234, 248, 258, 264, 272, 284, 296, 310, 320, 332, 324, 332, 344, 360, 350, 354, 350, 356, 342, 346, 354, 362, 372, 382, 392, 404, 416, 428, 440, 452, 464, 478, 492, 502, 514, 526, 504, 510, 516, 492

Offset

0,2

Comments

The index (also called "exponent" or "critical exponent") is the largest possible value of n/p, where n is the length of a subword (contiguous block) with period p.

A word is balanced if, for pairs of subwords of the same length, the number of 0's differ by at most 1.

It is known that there exist infinite balanced words with index (5+sqrt(5))/2; for example, the infinite Fibonacci word (A003849). So a(n) is positive for all n. Furthermore, (5 + sqrt(5))/2 is the minimum possible index for which there exist balanced infinite words, which accounts for its special role here.

Links

Table of n, a(n) for n=0..69.

Example

For n=5 the 18 words are {00010, 00100, 00101, 01110, 01010, 01001} and their reversals and complements.

Crossrefs

Cf. A003849.

The balanced words (with no restriction on index) are counted by A005598.

Sequence in context: A007590 A080476 A256885 * A330130 A053799 A343949

Adjacent sequences: A293492 A293493 A293494 * A293496 A293497 A293498

Keyword

nonn

Author

Jeffrey Shallit, Oct 10 2017

Status

approved