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

%I A209400 #24 May 23 2025 14:27:46
%S A209400 0,0,0,0,2,7,19,46,107,242,535,1162,2490,5281,11108,23206,48206,99663,
%T A209400 205218,421115,861585,1758249,3580075,7275377,14759592,29897683,
%U A209400 60481359,122206014,246665382,497414751,1002231335,2017877779,4060069150,8164204342
%N A209400 Number of subsets of {1,...,n} containing a subset of the form {k,k+1,k+3} for some k.
%C A209400 Also, number of subsets of {1,...,n} containing {a,a+2,a+3} for some a.
%C A209400 Also, number of bitstrings of length n containing 1101 or 1111.
%H A209400 David Nacin, <a href="/A209400/b209400.txt">Table of n, a(n) for n = 0..500</a>
%H A209400 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-2,1,-1,-2).
%F A209400 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.
%F A209400 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)).
%F A209400 a(n) = 2^n - A164387(n).
%e A209400 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.
%t A209400 LinearRecurrence[{3, -1, -2, 1, -1, -2}, {0, 0, 0, 0, 2, 7}, 40]
%t A209400 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 *)
%o A209400 (Python)
%o A209400 #From recurrence
%o A209400 def a(n, adict={0:0, 1:0, 2:0, 3:0, 4:2, 5:7}):
%o A209400  if n in adict:
%o A209400   return adict[n]
%o A209400  adict[n]=3*a(n-1)-a(n-2)-2*a(n-3)+a(n-4)-a(n-5)-2*a(n-6)
%o A209400  return adict[n]
%o A209400 (Python)
%o A209400 #Returns the actual list of valid subsets
%o A209400 def contains1101(n):
%o A209400  patterns=list()
%o A209400  for start in range (1,n-2):
%o A209400   s=set()
%o A209400   for i in range(4):
%o A209400    if (1,1,0,1)[i]:
%o A209400     s.add(start+i)
%o A209400   patterns.append(s)
%o A209400  s=list()
%o A209400  for i in range(2,n+1):
%o A209400   for temptuple in comb(range(1,n+1),i):
%o A209400    tempset=set(temptuple)
%o A209400    for sub in patterns:
%o A209400     if sub <= tempset:
%o A209400      s.append(tempset)
%o A209400      break
%o A209400  return s
%o A209400 #Counts all such sets
%o A209400 def countcontains1101(n):
%o A209400  return len(contains1101(n))
%o A209400 (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
%o A209400 (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
%Y A209400 Cf. A209398, A209399, A164387.
%K A209400 nonn,easy
%O A209400 0,5
%A A209400 _David Nacin_, Mar 07 2012