This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A209398 Number of subsets of {1,...,n} containing two elements whose difference is 2. 4
 0, 0, 0, 2, 7, 17, 39, 88, 192, 408, 855, 1775, 3655, 7478, 15228, 30898, 62511, 126177, 254223, 511472, 1027840, 2063600, 4140015, 8300767, 16635087, 33324462, 66736764, 133615658, 267461287, 535294673, 1071191415, 2143357000, 4288290240, 8579130888 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Also, the number of bitstrings of length n containing either 101 or 111. LINKS David Nacin, Table of n, a(n) for n = 0..500 Index entries for linear recurrences with constant coefficients, signature (3,-2,1,-1,-2). FORMULA a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) - a(n-4) - 2*a(n-5), a(0)=0, a(1)=0, a(2)=0, a(3)=2, a(4)=7. a(n) = 2^n - F(2+floor(n/2))*F(floor(2+(n+1)/2)), where F(n) are the Fibonacci numbers. a(n) = 2^n - A006498(n+2). G.f.: (2*x^3 + 1*x^4)/(1 - 3*x + 2*x^2 - x^3 + x^4 + 2*x^5) = x^3*(2 + x) / ((1 - 2*x)*(1 + x^2)*(1 - x - x^2)). EXAMPLE For n=3 the subsets containing 1 and 3 are {1,3} and {1,2,3} so a(3)=2. MATHEMATICA Table[2^n -Fibonacci[Floor[n/2] + 2]*Fibonacci[Floor[(n + 1)/2] + 2], {n, 0, 30}] LinearRecurrence[{3, -2, 1, -1, -2}, {0, 0, 0, 2, 7}, 40] CoefficientList[ Series[x^3 (x +2)/(2x^5 +x^4 -x^3 +2x^2 -3x +1), {x, 0, 33}], x] (* Robert G. Wilson v, Jan 03 2018 *) a[n_] := Floor[ N[(2^-n ((50 - 14 Sqrt[5]) (1 - Sqrt[5])^n + ((-1 + 2I) (-2I)^n - (1 + 2I) (2I)^n + 5 4^n) (15 + 11 Sqrt[5]) - 2 (1 + Sqrt[5])^n (85 + 37 Sqrt[5])))/(150 + 110 Sqrt[5])]]; Array[a, 33] (* Robert G. Wilson v, Jan 03 2018 *) PROG (Python) #Through Recurrence def a(n, adict={0:0, 1:0, 2:0, 3:2, 4:7}): .if n in adict: ..return adict[n] .adict[n]=3*a(n-1)-2*a(n-2)+a(n-3)-a(n-4)-2*a(n-5) .return adict[n] (Python) #Returns the actual list of valid subsets def contains101(n): .patterns=list() .for start in range (1, n-1): ..s=set() ..for i in range(3): ...if (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 subsets using the preceding function def countcontains101(n): .return len(contains101(n)) (PARI) x='x+O('x^30); concat([0, 0, 0], Vec(x^3*(2+x)/((1-2*x)*(1+x^2)*(1-x-x^2)))) \\ G. C. Greubel, Jan 03 2018 (MAGMA) [2^n - Fibonacci(Floor(n/2) + 2)*Fibonacci(Floor((n + 1)/2) + 2): n in [0..30]]; // G. C. Greubel, Jan 03 2018 CROSSREFS Cf. A006498, A209399, A209400. Sequence in context: A154117 A173769 A067038 * A175660 A175120 A239357 Adjacent sequences:  A209395 A209396 A209397 * A209399 A209400 A209401 KEYWORD nonn,easy AUTHOR David Nacin, Mar 07 2012 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 24 19:06 EDT 2018. Contains 301215 sequences. (Running on oeis4.)