A209400 Number of subsets of {1,...,n} containing a subset of the form {k,k+1,k+3} for some k.

0, 0, 0, 0, 2, 7, 19, 46, 107, 242, 535, 1162, 2490, 5281, 11108, 23206, 48206, 99663, 205218, 421115, 861585, 1758249, 3580075, 7275377, 14759592, 29897683, 60481359, 122206014, 246665382, 497414751, 1002231335, 2017877779, 4060069150, 8164204342

Offset: 0

Author

David Nacin, Mar 07 2012

Keywords

• nonn
• easy

Comments

Also, number of subsets of {1,...,n} containing {a,a+2,a+3} for some a.

Also, number of bitstrings of length n containing 1101 or 1111.

Examples

			When n=4 the only subsets containing an {a,a+1,a+3} happen when a=1 with the two subsets {1,2,3,4} and {1,2,4}.  Thus a(4)=2.
		

Links

• David Nacin, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-2,1,-1,-2).

Crossrefs

Cf. A209398, A209399, A164387.

Programs

Magma

I:=[0, 0, 0, 0, 2, 7]; [n le 6 select I[n] else 3*Self(n-1) - Self(n-2)-2*Self(n-3)+Self(n-4)-Self(n-5)-2*Self(n-6): n in [0..30]]; // G. C. Greubel, Jan 03 2018

Mathematica

LinearRecurrence[{3, -1, -2, 1, -1, -2}, {0, 0, 0, 0, 2, 7}, 40]
CoefficientList[Series[x^4*(2+x)/(1-3*x+x^2+2*x^3-x^4+x^5+2*x^6), {x,0, 50}], x] (* G. C. Greubel, Jan 03 2018 *)

PARI

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

Python

#From recurrence
def a(n, adict={0:0, 1:0, 2:0, 3:0, 4:2, 5:7}):
 if n in adict:
  return adict[n]
 adict[n]=3*a(n-1)-a(n-2)-2*a(n-3)+a(n-4)-a(n-5)-2*a(n-6)
 return adict[n]

Python

#Returns the actual list of valid subsets
def contains1101(n):
 patterns=list()
 for start in range (1,n-2):
  s=set()
  for i in range(4):
   if (1,1,0,1)[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 countcontains1101(n):
 return len(contains1101(n))

Formula

a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) + a(n-4) - a(n-5) - 2*a(n-6), a(4)=2, a(5)=7, a(i)=0 for i<4.

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

a(n) = 2^n - A164387(n).