A208902 The sum over all bitstrings b of length n of the number of runs in b not immediately followed by a longer run.

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

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)).

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

Cf. A056453, A208900, A208901, A208903.

Sequence in context: A124614 A070933 A059570 * A018016 A182644 A099425

Adjacent sequences: A208899 A208900 A208901 * A208903 A208904 A208905

Keyword

nonn,easy

Author

David Nacin, Mar 03 2012

Status

approved