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A234933 The number of binary sequences that contain at least two consecutive 1's and contain at least two consecutive 0's. 2
0, 0, 0, 0, 2, 8, 24, 62, 148, 336, 738, 1584, 3344, 6974, 14412, 29576, 60370, 122712, 248616, 502398, 1013156, 2039840, 4101570, 8238560, 16534432, 33161598, 66473244, 133189272, 266771378, 534178376, 1069385208, 2140434494, 4283561524, 8571479664, 17150008482, 34311422736, 68641300400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-4,-1,2).

FORMULA

a(n) = 2*A232580(n-1) for n>0.

G.f.: 2*x^4/(1 - 4*x + 4*x^2 + x^3 - 2*x^4).

From Colin Barker, Nov 03 2016: (Start)

a(n) = 2^(-n)*(5*2^n*(2+2^n)+(1-sqrt(5))^n*(-5+3*sqrt(5))-(1+sqrt(5))^n*(5+3*sqrt(5)))/5 for n>0.

a(n) = 4*a(n-1)-4*a(n-2)-a(n-3)+2*a(n-4) for n>4.

(End)

a(n) = 2*(A000079(n-1)-A000045(n+2)+1) for n>0. - Ehren Metcalfe, Dec 27 2018

EXAMPLE

a(5) = 8 because we have:

1: {0, 0, 0, 1, 1},

2: {0, 0, 1, 1, 0},

3: {0, 0, 1, 1, 1},

4: {0, 1, 1, 0, 0},

5: {1, 0, 0, 1, 1},

6: {1, 1, 0, 0, 0},

7: {1, 1, 0, 0, 1},

8: {1, 1, 1, 0, 0}.

MATHEMATICA

nn = 25; a = (x + x^2)/(1 - x^2); b = 1/(1 - 2x); c = 1/(1 - x - x^2); CoefficientList[Series[2x^3 a b c, {x, 0, nn}], x]

(* or *)

Table[Length[Select[Tuples[{0, 1}, n], MatchQ[#, {___, 1, 1, ___}] && MatchQ[#, {___, 0, 0, ___}] &]], {n, 0, 15}]

Join[{0}, LinearRecurrence[{4, -4, -1, 2}, {0, 0, 0, 2}, 40]] (* Vincenzo Librandi, Dec 28 2018 *)

PROG

(PARI) concat([0, 0, 0, 0], Vec(2*x^4/(1-4*x+4*x^2+x^3-2*x^4)+O(x^66))) \\ Joerg Arndt, Jan 04 2014

(MAGMA) I:=[0, 0, 0, 0, 2]; [n le 5 select I[n] else 4*Self(n-1)-4*Self(n-2)-Self(n-3)+2*Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 28 2018

CROSSREFS

Cf. A000045, A000079, A232580.

Sequence in context: A263598 A075218 A006728 * A222808 A075216 A127790

Adjacent sequences:  A234930 A234931 A234932 * A234934 A234935 A234936

KEYWORD

nonn,easy

AUTHOR

Geoffrey Critzer, Jan 01 2014

STATUS

approved

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Last modified July 8 19:01 EDT 2020. Contains 335524 sequences. (Running on oeis4.)