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 A118870 Number of binary sequences of length n with no subsequence 0101. 6
 1, 2, 4, 8, 15, 28, 53, 100, 188, 354, 667, 1256, 2365, 4454, 8388, 15796, 29747, 56020, 105497, 198672, 374140, 704582, 1326871, 2498768, 4705689, 8861770, 16688516, 31427872, 59185079, 111457548, 209897245, 395279228, 744391228, 1401840170 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Column 0 of A118869 and column 10 of A209972. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-1). FORMULA G.f.: (1 +x^2)/(1 -2*x +x^2 -2*x^3 +x^4). a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - a(n-4) for n>=4. a(n) = A112575(n-1) + A112575(n+1). - R. J. Mathar, Dec 10 2011 EXAMPLE a(5) = 28 because among the 32 (=2^5) binary sequences of length 5 only 01010, 01011, 00101 and 10101 contain the subsequence 0101. MAPLE a[0]:=1:a[1]:=2:a[2]:=4:a[3]:=8: for n from 4 to 35 do a[n]:=2*a[n-1]-a[n-2]+2*a[n-3]-a[n-4] od: seq(a[n], n=0..35); MATHEMATICA CoefficientList[Series[(1+x^2)/(1-2x+x^2-2x^3+x^4), {x, 0, 40}], x] (* Geoffrey Critzer, Nov 28 2013 *) PROG (Magma) [n le 4 select 2^(n-1) else 2*Self(n-1) -Self(n-2) +2*Self(n-3) -Self(n-4): n in [1..41]]; // G. C. Greubel, Jan 14 2022 (Sage) @CachedFunction def A112575(n): return sum((-1)^k*binomial(n-k, k)*lucas_number1(n-2*k, 2, -1) for k in (0..(n/2))) def A118870(n): return A112575(n-1) + A112575(n+1) [A118870(n) for n in (0..40)] # G. C. Greubel, Jan 14 2022 CROSSREFS Cf. A000071, A000073, A005251, A112575, A118869, A209972, A332052. Sequence in context: A049864 A239554 A268393 * A171857 A190160 A332052 Adjacent sequences: A118867 A118868 A118869 * A118871 A118872 A118873 KEYWORD nonn AUTHOR Emeric Deutsch, May 03 2006 STATUS approved

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Last modified May 29 01:48 EDT 2023. Contains 363029 sequences. (Running on oeis4.)