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

1,1

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 (3,0,-6,4).

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

Example

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

Mathematica

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

Crossrefs

Cf. A056453, A208901, A208902, A208903.

Sequence in context: A284462 A262278 A265639 * A229733 A051765 A265987

Adjacent sequences: A208897 A208898 A208899 * A208901 A208902 A208903

Keyword

nonn,easy

Author

David Nacin, Mar 03 2012

Status

approved