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
KEYWORD
nonn,easy
AUTHOR
David Nacin, Mar 08 2012
STATUS
approved