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A209399 Number of subsets of {1,...,n} containing two elements whose difference is 3. 3
0, 0, 0, 0, 4, 14, 37, 83, 181, 387, 824, 1728, 3584, 7360, 15032, 30571, 61987, 125339, 252883, 509294, 1024300, 2057848, 4130724, 8285758, 16610841, 33285207, 66673209, 133512759, 267294832, 535025408, 1070755840, 2142652160, 4287149680, 8577285255 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Also, the number of bitstrings of length n containing one of the following: 1001, 1101, 1011, 1111.

LINKS

David Nacin, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,3,-1,-1,-3,1,2).

FORMULA

a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 3*a(n-4) - a(n-5) - a(n-6) - 3*a(n-7) + a(n-8) + 2*a(n-9).

G.f.: (4*x^4 + 2*x^5 - x^6 - 2*x^7 - x^8)/(1 - 3*x + 1*x^2 + 3*x^3 - 3*x^4 + x^5 + x^6 + 3*x^7 - x^8 - 2*x^9) = x^4*(4 + 2*x - x^2 - 2*x^3 - x^4)/((1 - 2*x)*(1 - x - x^2)*(1 + x^3 - x^6)).

a(n) = 2^n - A006500(n).

a(n) = 2^n - Product(i=0 to 2) F(floor((n+i)/3)+2) where F(n) is the n-th Fibonacci number.

EXAMPLE

When n=4 any such subset must contain 1 and 4.  There are four such subsets so a(4) = 4.

MATHEMATICA

LinearRecurrence[{3, -1, -3, 3, -1, -1, -3, 1, 2}, {0, 0, 0, 0, 4, 14, 37, 83, 181}, 50]

Table[2^n - Product[Fibonacci[Floor[(n + i)/3] + 2], {i, 0, 2}], {n, 0, 50}]

PROG

(Python)

#From recurrence

def a(n, adict={0:0, 1:0, 2:0, 3:0, 4:4, 5:14, 6:37, 7:83, 8:181}):

.if n in adict:

..return adict[n]

.adict[n]=3*a(n-1)-a(n-2)-3*a(n-3)+3*a(n-4)-a(n-5)-a(n-6)-3*a(n-7)+a(n-8)+2*a(n-9)

.return adict[n]

(Python)

#Returns the actual list of valid subsets

def contains1001(n):

.patterns=list()

.for start in range (1, n-2):

..s=set()

..for i in range(4):

...if (1, 0, 0, 1)[i]:

....s.add(start+i)

..patterns.append(s)

.s=list()

.for i in range(2, n+1):

..for temptuple in comb(range(1, n+1), i):

...tempset=set(temptuple)

...for sub in patterns:

....if sub <= tempset:

.....s.append(tempset)

.....break

.return s

#Counts all such sets

def countcontains1001(n):

.return len(contains1001(n))

(PARI) x='x+O('x^30); concat([0, 0, 0, 0], Vec(x^4*(4+2*x-x^2-2*x^3-x^4)/( (1-2*x)*(1-x-x^2)*(1+x^3-x^6)))) \\ G. C. Greubel, Jan 03 2018

(MAGMA) [2^n - Fibonacci(Floor(n/3) + 2)*Fibonacci(Floor((n + 1)/3) + 2)*Fibonacci(Floor((n + 2)/3) + 2): n in [0..30]]; // G. C. Greubel, Jan 03 2018

CROSSREFS

Cf. A209398, A209400, A006500.

Sequence in context: A317148 A027166 A126943 * A192974 A187428 A316878

Adjacent sequences:  A209396 A209397 A209398 * A209400 A209401 A209402

KEYWORD

nonn,easy

AUTHOR

David Nacin, Mar 07 2012

STATUS

approved

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Last modified November 25 19:43 EST 2020. Contains 338625 sequences. (Running on oeis4.)