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A208900 Number of bitstrings of length n which (if having two or more runs) the last two runs have different lengths. 5
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 (list; graph; refs; listen; history; text; internal format)
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)).

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 A265987 A076907

Adjacent sequences:  A208897 A208898 A208899 * A208901 A208902 A208903

KEYWORD

nonn,easy

AUTHOR

David Nacin, Mar 03 2012

STATUS

approved

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Last modified September 26 12:04 EDT 2021. Contains 347665 sequences. (Running on oeis4.)