A209400 - OEIS

%I #18 Sep 08 2022 08:46:01

%S 0,0,0,0,2,7,19,46,107,242,535,1162,2490,5281,11108,23206,48206,99663,

%T 205218,421115,861585,1758249,3580075,7275377,14759592,29897683,

%U 60481359,122206014,246665382,497414751,1002231335,2017877779,4060069150,8164204342

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

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

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

%H David Nacin, <a href="/A209400/b209400.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-2,1,-1,-2).

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

%e 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 LinearRecurrence[{3, -1, -2, 1, -1, -2}, {0, 0, 0, 0, 2, 7}, 40]

%t 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 (Python)

%o #From recurrence

%o def a(n, adict={0:0, 1:0, 2:0, 3:0, 4:2, 5:7}):

%o .if n in adict:

%o ..return adict[n]

%o .adict[n]=3*a(n-1)-a(n-2)-2*a(n-3)+a(n-4)-a(n-5)-2*a(n-6)

%o .return adict[n]

%o (Python)

%o #Returns the actual list of valid subsets

%o def contains1101(n):

%o .patterns=list()

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

%o ..s=set()

%o ..for i in range(4):

%o ...if (1,1,0,1)[i]:

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

%o ..patterns.append(s)

%o .s=list()

%o .for i in range(2,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 .....s.append(tempset)

%o .....break

%o .return s

%o #Counts all such sets

%o def countcontains1101(n):

%o .return len(contains1101(n))

%o (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 (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]]; //

%o _G. C. Greubel_, Jan 03 2018

%Y Cf. A209398, A209399, A164387

%K nonn,easy

%O 0,5

%A _David Nacin_, Mar 07 2012