A209410 - OEIS

%I #14 Sep 08 2022 08:46:01

%S 1,2,4,8,16,28,49,91,169,312,576,1056,1936,3564,6561,12069,22201,

%T 40826,75076,138096,254016,467208,859329,1580535,2907025,5346880,

%U 9834496,18088448,33269824,61192712,112550881,207013417,380757169,700321570,1288092100

%N Number of subsets of {1,...,n} not containing {a,a+2,a+4} for any a.

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

%H G. C. Greubel, <a href="/A209410/b209410.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 2, 0, 2, 2, 0, -1, -1).

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

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

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

%F 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)).

%e 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.

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

%o (Python)

%o #Returns the actual list of valid subsets

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

%o .patterns=list()

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

%o ..s=set()

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

%o ...if bitstring[i]:

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

%o ..patterns.append(s)

%o .s=list()

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

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

%o ...s.append(smallset)

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

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

%o ...tempset=set(temptuple)

%o ...for sub in patterns:

%o ....if sub <= tempset:

%o .....status=False

%o .....break

%o ...if status:

%o ....s.append(tempset)

%o .return s

%o #Counts all such sets

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

%o .return len(avoidscode(n))

%o #From recurrence

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

%o .if n in adict:

%o ..return adict[n]

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

%o .return adict[n]

%o (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

%o (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

%Y Cf. A209408, A209409

%K nonn,easy

%O 0,2

%A _David Nacin_, Mar 08 2012