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A209410 Number of subsets of {1,...,n} not containing {a,a+2,a+4} for any a. 3
1, 2, 4, 8, 16, 28, 49, 91, 169, 312, 576, 1056, 1936, 3564, 6561, 12069, 22201, 40826, 75076, 138096, 254016, 467208, 859329, 1580535, 2907025, 5346880, 9834496, 18088448, 33269824, 61192712, 112550881, 207013417, 380757169, 700321570, 1288092100 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Also, the number of bitstrings of length n not containing 10101,11101,10111 or 11111.

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (1, 0, 2, 0, 2, 2, 0, -1, -1).

FORMULA

a(n) = 2^n - A209409(n).

a(n) = t[floor(n/2)+2]*t[floor((n+1)/2)+2] where t(n) is the n-th tribonacci number.

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

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

EXAMPLE

For n=5, subsets containing {a,a+2,a+4} occur only when a=1.  There are 2^2 subsets including {1,3,5}, thus a(5) = 2^5 - 4 = 28.

MATHEMATICA

LinearRecurrence[{1, 0, 2, 0, 2, 2, 0, -1, -1}, {1, 2, 4, 8, 16, 28, 49, 91, 169}, 40]

PROG

(Python)

#Returns the actual list of valid subsets

def avoidscode(n, bitstring=(1, 0, 1, 0, 1)):

.patterns=list()

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

..s=set()

..for i in range(len(bitstring)):

...if bitstring[i]:

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

..patterns.append(s)

.s=list()

.for i in range(sum(bitstring)):

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

...s.append(smallset)

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

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

...tempset=set(temptuple)

...for sub in patterns:

....if sub <= tempset:

.....status=False

.....break

...if status:

....s.append(tempset)

.return s

#Counts all such sets

def countavoidscode(n, bitstring=(1, 0, 1, 0, 1)):

.return len(avoidscode(n))

#From recurrence

def a(n, adict={0:1, 1:2, 2:4, 3:8, 4:16, 5:28, 6:49, 7:91, 8:169}):

.if n in adict:

..return adict[n]

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

.return adict[n]

(PARI) x='x+O('x^30); Vec((1+x+2*x^2+2*x^3+4*x^4+2*x^5-x^6-2*x^7-x^8)/( (-1+x+x^2+x^3)*(-1-x^2-x^4+x^6))) \\ G. C. Greubel, Jan 05 2018

(MAGMA) I:=[1, 2, 4, 8, 16, 28, 49, 91, 169]; [n le 9 select I[n] else Self(n-1)+2*Self(n-3)+2*Self(n-5)+2*Self(n-6)-Self(n-8)-Self(n-9): n in [1..30]]; // G. C. Greubel, Jan 05 2018

CROSSREFS

Cf. A209408, A209409

Sequence in context: A208933 A227036 A172020 * A228733 A318767 A208531

Adjacent sequences:  A209407 A209408 A209409 * A209411 A209412 A209413

KEYWORD

nonn,easy

AUTHOR

David Nacin, Mar 08 2012

STATUS

approved

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Last modified August 15 16:02 EDT 2020. Contains 336505 sequences. (Running on oeis4.)