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 A065455 Number of (binary) bit strings of length n in which no even block of 0's is followed by an odd block of 1's. 8
 1, 2, 4, 7, 14, 25, 49, 89, 172, 316, 605, 1120, 2131, 3965, 7513, 14026, 26504, 49591, 93538, 175277, 330205, 619369, 1165892, 2188312, 4117045, 7730828, 14539447, 27309529, 51349169, 96468034, 181357036, 340753271, 640539142, 1203616849 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The limit of the ratio of successive terms as n increases can be shown to be 2*cos(Pi/9). In the opposite direction, as n—->-oo (see A052545), a(n+1)/a(n) approaches 2*cos(5*Pi/9). For example, a(-6)/a(-7) = -92/265, which is close to 2*cos(5*Pi/9). - Richard Locke Peterson, Apr 22 2019 Let P(n, j, m) = Sum_{r=1..m} (2^n*(1-(-1)^r)*cos(Pi*r/(m+1))^n*cot(Pi*r/(2*(m+1)))* sin(j*Pi*r/(m+1)))/(m+1) denote the number of paths of length n starting at the j-th node on the path graph P_m. We have a(n) = P(n, 3, 8). - Herbert Kociemba, Sep 17 2020 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (0,3,1). FORMULA G.f.: (1+x)^2/(1-3*x^2-x^3). EXAMPLE a(5) = 32-7 = 25 because 00111, 00101, 00100, 10010, 01001, 11001, 00001 are forbidden. MATHEMATICA LinearRecurrence[{0, 3, 1}, {1, 2, 4}, 40] (* G. C. Greubel, May 31 2019 *) a[n_, j_, m_]:=Sum[(2^(n+1)Cos[Pi r/(m+1)]^n Cot[Pi r/(2(m+1))] Sin[j Pi r/(m+1)])/(m+1), {r, 1, m, 2}] Table[a[n, 3, 8], {n, 0, 40}]//Round (* Herbert Kociemba, Sep 17 2020 *) CoefficientList[Series[(1+x)^2/(1-3x^2-x^3), {x, 0, 50}], x] (* Harvey P. Dale, Jul 16 2021 *) PROG (PARI) a(n)=([0, 1, 0; 0, 0, 1; 1, 3, 0]^n*[1; 2; 4])[1, 1] \\ Charles R Greathouse IV, Jun 11 2015 (Magma) I:=[1, 2, 4]; [n le 3 select I[n] else 3*Self(n-2) +Self(n-3): n in [1..40]]; // G. C. Greubel, May 31 2019 (Sage) ((1+x)^2/(1-3*x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 31 2019 (GAP) a:=[1, 2, 4];; for n in [4..40] do a[n]:=3*a[n-2]+a[n-3]; od; a; # G. C. Greubel, May 31 2019 CROSSREFS Cf. A061279 (forbids odd block 0's-odd block 1's), A065494, A065495, A065497. Cf. A052545 (this is what we get if n takes negative values). Sequence in context: A065491 A072810 A167606 * A220842 A026010 A088813 Adjacent sequences: A065452 A065453 A065454 * A065456 A065457 A065458 KEYWORD nonn,easy AUTHOR Len Smiley, Nov 24 2001 STATUS approved

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Last modified September 13 08:10 EDT 2024. Contains 375898 sequences. (Running on oeis4.)