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

Author

David Nacin, Mar 03 2012

Keywords

• nonn
• easy

Comments

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

Examples

			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.
		

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).

Crossrefs

Cf. A056453, A208900, A208902, A208903.

Programs

Mathematica

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

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