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A209408 Number of subsets of {1,...,n} containing {a,a+4} for some a. 3
0, 0, 0, 0, 0, 8, 28, 74, 175, 377, 799, 1673, 3471, 7192, 14784, 30208, 61440, 124416, 251328, 506712, 1020015, 2051015, 4119775, 8268215, 16582735, 33239558, 66599068, 133392344, 267099120, 534709192, 1070244924, 2141826898, 4285816671, 8575127217 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

For n=5, subsets containing {a,a+4} occur only when a=1.  There are 2^3 subsets including {1,5}, thus a(5) = 8.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-1,-2,-2,6,-2,-4,2,-6,2,4,1,-3, 1,2).

FORMULA

a(n) = 2^n - A208741(n-1).

a(n) = 2^n - Product_{i=0..3} Fibonacci(floor((n + i)/4) + 2).

a(n) = 3*a(n-1) - a(n-2) -2*a(n-3) -2*a(n-4) + 6*a(n-5) - 2*a(n-6) - 4*a(n-7) + 2*a(n-8) - 6*a(n-9) + 2*a(n-10) + 4*a(n-11) + a(n-12) - 3*a(n-13) + a(n-14) + 2*a(n-15).

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

MATHEMATICA

Table[2^n - Product[Fibonacci[Floor[(n + i)/4] + 2], {i, 0, 3}], {n, 0, 30}]

LinearRecurrence[{3, -1, -2, -2, 6, -2, -4, 2, -6, 2, 4, 1, -3, 1, 2}, {0, 0, 0, 0, 0, 8, 28, 74, 175, 377, 799, 1673, 3471, 7192, 14784}, 30]

PROG

(Python)

#Returns the actual list of valid subsets

def contains10001(n):

.patterns=list()

.for start in range (1, n-3):

..s=set()

..for i in range(5):

...if (1, 0, 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 countcontains10001(n):

.return len(contains10001(n))

#From recurrence

def a(n, adict={0:0, 1:0, 2:0, 3:0, 4:0, 5:8, 6:28, 7:74, 8:175, 9:377, 10:799, 11:1673, 12:3471, 13:7192, 14:14784}):

.if n in adict:

..return adict[n]

.adict[n]=3*a(n-1)-a(n-2)-2*a(n-3)-2*a(n-4)+6*a(n-5)-2*a(n-6)-4*a(n-7)+2*a(n-8)-6*a(n-9)+2*a(n-10)+4*a(n-11)+a(n-12)-3*a(n-13)+a(n-14)+2*a(n-15)

.return adict[n]

(PARI) for(n=0, 20, print1(2^n - fibonacci(floor(n/4) + 2)*fibonacci( floor((n + 1)/4) + 2)*fibonacci(floor((n + 2)/4) + 2)*fibonacci( floor((n + 3)/4) + 2), ", ")) \\ G. C. Greubel, Jan 03 2018

(MAGMA) [2^n - Fibonacci(Floor(n/4) + 2)*Fibonacci(Floor((n + 1)/4) + 2)*Fibonacci(Floor((n + 2)/4) + 2)*Fibonacci(Floor((n + 3)/4) + 2): n in [0..30]]; // G. C. Greubel, Jan 03 2018

CROSSREFS

Cf. A031923, A209409, A209410.

Sequence in context: A105636 A102665 A212565 * A316214 A134638 A293289

Adjacent sequences:  A209405 A209406 A209407 * A209409 A209410 A209411

KEYWORD

nonn,easy

AUTHOR

David Nacin, Mar 08 2012

STATUS

approved

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Last modified August 13 19:30 EDT 2020. Contains 336451 sequences. (Running on oeis4.)