A208903 The sum over all bitstrings b of length n with at least two runs of the number of runs in b not immediately followed by a longer run.

0, 4, 12, 32, 76, 180, 412, 940, 2108, 4700, 10364, 22716, 49404, 106876, 229884, 492284, 1049596, 2229756, 4720636, 9964540, 20975612, 44046332, 92282876, 192950268, 402669564, 838885372, 1744863228, 3623927804, 7516258300, 15569354748, 32212385788

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

Cf. A056453, A208900, A208901, A208902.

Sequence in context: A001934 A004403 A084566 * A079769 A107035 A260145

Adjacent sequences: A208900 A208901 A208902 * A208904 A208905 A208906

Keyword

nonn,easy

Author

David Nacin, Mar 03 2012

Status

approved