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A208901
Number of bitstrings of length n (with at least two runs) where the last two runs have different lengths.
5
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
Aruna Gabhe, Problem 11623, Am. Math. Monthly 119 (2012) 161.
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
KEYWORD
nonn,easy
AUTHOR
David Nacin, Mar 03 2012
STATUS
approved