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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. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 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. 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). 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). EXAMPLE 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. 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 *) PROG (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)) (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 (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 CROSSREFS Cf. A209398, A209399, A164387 Sequence in context: A220697 A078842 A110299 * A112304 A006589 A238914 Adjacent sequences:  A209397 A209398 A209399 * A209401 A209402 A209403 KEYWORD nonn,easy AUTHOR David Nacin, Mar 07 2012 STATUS approved

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Last modified May 29 16:43 EDT 2020. Contains 334704 sequences. (Running on oeis4.)