A208900 Number of bitstrings of length n which (if having two or more runs) the last two runs have different lengths.

2, 2, 6, 10, 26, 50, 114, 226, 482, 962, 1986, 3970, 8066, 16130, 32514, 65026, 130562, 261122, 523266, 1046530, 2095106, 4190210, 8384514, 16769026, 33546242, 67092482, 134201346, 268402690, 536838146, 1073676290, 2147418114, 4294836226, 8589803522

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 where the last runs have different lengths are 111,000,100,011,110,001 so a(3) = 6.
		

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 (3,0,-6,4).

Crossrefs

Cf. A056453, A208901, A208902, A208903.

Programs

Mathematica

Table[2 + 2^n - 2^(Floor[n/2] + 1) ,  {n, 1, 40}]
LinearRecurrence[{3, 0, -6, 4}, {2, 2, 6, 10}, 40]

Formula

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

a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4), a(0) = 2, a(1) = 2, a(2) = 6, a(3) = 10.

G.f.: x*(2 - 4*x + 4*x^3)/((1-x)*(1-2*x^2)*(1-2*x)).

E.g.f.: - 2*cosh(sqrt(2)*x) + 2*exp(x)*(1 + sinh(x)) - sqrt(2)*sinh(sqrt(2)*x). - Stefano Spezia, Jun 06 2023