OFFSET
1,2
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 (5,-6,-6,16,-8).
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)) - 2.
a(n) = A208902(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) = 0, a(2) = 4, a(3) = 12, a(4) = 32, a(5) = 76.
G.f.: (4*x - 8*x^2 - 4*x^3 + 12*x^4)/(1 - 5*x + 6*x^2 + 6*x^3 - 16*x^4 +
8*x^5).
EXAMPLE
n=3: 101, 010 each have 3; 100, 011 each have 1; 001, 110 each have 2. (000, 111 do not have at least two runs so they do not contribute.) Summing these gives 6+2+4 = 12 so a(3) = 12.
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)) - 2, {n, 1, 40}]
LinearRecurrence[{5, -6, -6, 16, -8}, {0, 4, 12, 32, 76}, 40]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
David Nacin, Mar 03 2012
STATUS
approved