

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
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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 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(n1)  6*a(n2)  6*a(n3) + 16*a(n4)  8*a(n5), 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 + (n1)/2  (1/2)^(n1)  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, A208900A208902.
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



