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 A209409 Number of subsets of {1,...,n} containing {a,a+2,a+4} for some a. 3
 0, 0, 0, 0, 0, 4, 15, 37, 87, 200, 448, 992, 2160, 4628, 9823, 20699, 43335, 90246, 187068, 386192, 794560, 1629944, 3334975, 6808073, 13870191, 28207552, 57274368, 116129280, 235165632, 475678200, 961190943, 1940470231, 3914210127, 7889613022, 15891777084 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Also, the number of bitstrings of length n containing 10101,11101,10111 or 11111. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (3,-2,2,-4,2,-2,-4,-1,1,2). FORMULA A(n) = 2^n - A209410(n) a(n) = 2^n - t[floor(n/2)+2]*t[floor((n+1)/2)+2] where t(n) is the n-th tribonacci number. a(n) = 3*a(n-1) - 2*a(n-2) + 2*a(n-3) - 4*a(n-4) + 2*a(n-5) - 2*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) + 2*a(n-10). G.f.: x^5*(4 + 3*x - 2*x^3 - x^4)/((1 - 2*x) (1 - x - x^2 - x^3) (1 + x^2 + x^4 - x^6)). EXAMPLE For n=5, subsets containing {a,a+2,a+4} occur only when a=1.  There are 2^2 subsets including {1,3,5}, thus a(5) = 4. MATHEMATICA LinearRecurrence[{3, -2, 2, -4, 2, -2, -4, -1, 1, 2}, {0, 0, 0, 0, 0, 4, 15, 37, 87, 200}, 40] PROG (Python) #Returns the actual list of valid subsets def containscode(n, bitstring=(1, 0, 1, 0, 1)): .patterns=list() .for start in range (1, n-len(bitstring)+2): ..s=set() ..for i in range(len(bitstring)): ...if bitstring[i]: ....s.add(start+i) ..patterns.append(s) .s=list() .for i in range(sum(bitstring), 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 countcontainscode(n, bitstring=(1, 0, 1, 0, 1)): .return len(containscode(n)) (Python) #From recurrence def a(n, adict={0:0, 1:0, 2:0, 3:0, 4:0, 5:4, 6:15, 7:37, 8:87, 9:200}): .if n in adict: ..return adict[n] .adict[n]=3*a(n-1) - 2*a(n-2) + 2*a(n-3) - 4*a(n-4) + 2*a(n-5) - 2*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) + 2*a(n-10) .return adict[n] (PARI) x='x+O('x^30); concat([0, 0, 0, 0, 0], Vec(x^5*(4+3*x-2*x^3-x^4)/((1- 2*x)*(1-x-x^2-x^3)*(1+x^2+x^4-x^6)))) \\ G. C. Greubel, Jan 03 2018 CROSSREFS Cf. A209408, A209410. Sequence in context: A033813 A296295 A296268 * A241302 A112666 A014629 Adjacent sequences:  A209406 A209407 A209408 * A209410 A209411 A209412 KEYWORD nonn,easy AUTHOR David Nacin, Mar 08 2012 STATUS approved

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Last modified August 11 18:37 EDT 2020. Contains 336428 sequences. (Running on oeis4.)