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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. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A run is a maximal subsequence of (possibly just one) identical bits.

REFERENCES

A. Gabhe, Problems and Solutions: 11623, The Amer. Math. Monthly 119 (2012), no. 2, 161.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index to sequences with 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

When 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. (000,111 do not have at least two runs so they do not contribute.

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-A208902.

Sequence in context: A001934 A004403 A084566 * A079769 A107035 A118885

Adjacent sequences:  A208900 A208901 A208902 * A208904 A208905 A208906

KEYWORD

nonn,easy

AUTHOR

David Nacin, Mar 03 2012

STATUS

approved

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Last modified December 19 15:15 EST 2014. Contains 252232 sequences.