A208901 Number of bitstrings of length n (with at least two runs) where the last two runs have different lengths.

0, 0, 4, 8, 24, 48, 112, 224, 480, 960, 1984, 3968, 8064, 16128, 32512, 65024, 130560, 261120, 523264, 1046528, 2095104, 4190208, 8384512, 16769024, 33546240, 67092480, 134201344, 268402688, 536838144, 1073676288, 2147418112, 4294836224, 8589803520

Offset

1,3

Comments

A run is a maximal subsequence of (possibly just one) identical bits.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Aruna Gabhe, Problem 11623, Am. Math. Monthly 119 (2012) 161.

Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Formula

a(n) = 2^n - 2^(floor(n/2)+1).

a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3), a(0) = 0, a(1) = 0, a(2) = 4.

G.f.: 4*x^2/((1 - 2*x)*(1 - 2*x^2)).

E.g.f.: 2*(cosh(2*x) - cosh(sqrt(2)*x) + sinh(2*x) - sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, Jun 06 2023

Example

If n=3 the bitstrings (with at least two runs) where the last runs have different lengths are 100,011,110,001 so a(3) = 4.

Mathematica

Table[2^n - 2^(Floor[ n/2] + 1) ,  {n, 1, 40}]
LinearRecurrence[{2, 2, -4}, {0, 0, 4}, 40]

Crossrefs

Cf. A056453, A208900, A208902, A208903.

Sequence in context: A180002 A266821 A306484 * A319721 A115641 A153334

Adjacent sequences: A208898 A208899 A208900 * A208902 A208903 A208904

Keyword

nonn,easy

Author

David Nacin, Mar 03 2012

Status

approved