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A208902
The sum over all bitstrings b of length n of the number of runs in b not immediately followed by a longer run.
5
2, 6, 14, 34, 78, 182, 414, 942, 2110, 4702, 10366, 22718, 49406, 106878, 229886, 492286, 1049598, 2229758, 4720638, 9964542, 20975614, 44046334, 92282878, 192950270, 402669566, 838885374, 1744863230, 3623927806, 7516258302, 15569354750, 32212385790
OFFSET
1,1
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 + (n - 1)/2 - (1/2)^(n - 1) - 2 (1 - (1/2)^floor(n/2)) + (1/2)^(floor(n/2) + 1) (1 + (-1)^n)).
a(n) = A208903(n) + 2.
a(n) = 5*a(n-1) - 6*a(n-2) - 6*a(n-3) + 16*a(n-4) - 8*a(n-5), a(1) = 2, a(2) = 6, a(3) = 14, a(4) = 34, a(5) = 78.
G.f.: (2 - 4*x - 4*x^2 + 12*x^3 - 4*x^4)/(1 - 5*x + 6*x^2 + 6*x^3 - 16*x^4 + 8*x^5).
EXAMPLE
When n=3, 000,111 each have 1 such run, 101,010 each have 3, 100,011 each have 1, 001, 110 each have 2, summing these gives 2+6+2+4=14 so a(3) = 14.
MATHEMATICA
Table[2^n*(2 + (n - 1)/2 - (1/2)^(n - 1) - 2*(1 - (1/2)^Floor[n/2]) + (1/2)^(Floor[n/2] + 1) (1 + (-1)^n)), {n, 1, 40}]
LinearRecurrence[{5, -6, -6, 16, -8}, {2, 6, 14, 34, 78}, 40]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
David Nacin, Mar 03 2012
STATUS
approved