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A209409 Number of subsets of {1,...,n} containing {a,a+2,a+4} for some a. 3
0, 0, 0, 0, 0, 4, 15, 37, 87, 200, 448, 992, 2160, 4628, 9823, 20699, 43335, 90246, 187068, 386192, 794560, 1629944, 3334975, 6808073, 13870191, 28207552, 57274368, 116129280, 235165632, 475678200, 961190943, 1940470231, 3914210127, 7889613022, 15891777084 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Also, the number of bitstrings of length n 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 (3,-2,2,-4,2,-2,-4,-1,1,2).

FORMULA

A(n) = 2^n - A209410(n)

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

a(n) = 3*a(n-1) - 2*a(n-2) + 2*a(n-3) - 4*a(n-4) + 2*a(n-5) - 2*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) + 2*a(n-10).

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

MATHEMATICA

LinearRecurrence[{3, -2, 2, -4, 2, -2, -4, -1, 1, 2}, {0, 0, 0, 0, 0, 4, 15, 37, 87, 200}, 40]

PROG

(Python)

#Returns the actual list of valid subsets

def containscode(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), 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 countcontainscode(n, bitstring=(1, 0, 1, 0, 1)):

.return len(containscode(n))

(Python)

#From recurrence

def a(n, adict={0:0, 1:0, 2:0, 3:0, 4:0, 5:4, 6:15, 7:37, 8:87, 9:200}):

.if n in adict:

..return adict[n]

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

.return adict[n]

(PARI) x='x+O('x^30); concat([0, 0, 0, 0, 0], Vec(x^5*(4+3*x-2*x^3-x^4)/((1- 2*x)*(1-x-x^2-x^3)*(1+x^2+x^4-x^6)))) \\ G. C. Greubel, Jan 03 2018

CROSSREFS

Cf. A209408, A209410.

Sequence in context: A033813 A296295 A296268 * A241302 A112666 A014629

Adjacent sequences:  A209406 A209407 A209408 * A209410 A209411 A209412

KEYWORD

nonn,easy

AUTHOR

David Nacin, Mar 08 2012

STATUS

approved

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Last modified August 11 18:37 EDT 2020. Contains 336428 sequences. (Running on oeis4.)