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 A163733 Number of n X 2 binary arrays with all 1's connected, all corners 1, and no 1 having more than two 1's adjacent. 5
 1, 1, 2, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Same recurrence for A163695. Same recurrence for A163714. Appears to coincide with diagonal sums of A072405. [Paul Barry, Aug 10 2009] From Gary W. Adamson, Sep 15 2016: (Start) Let the sequence prefaced with a 1: (1, 1, 1, 2, 2, 4, 6,...) equate to r(x).  Then (r(x) * r(x^2) * r(x^4) * r(x^8) * ...) = the Fibonacci sequence, (1, 1, 2, 3, 5,...). Let M = the following production matrix: 1, 0, 0, 0, 0,... 1, 0, 0, 0, 0,... 1, 1, 0, 0, 0,... 2, 1, 0, 0, 0,... 2, 1, 1, 0, 0,... 4, 2, 1, 0, 0,... 6, 2, 1, 1, 0,... ... Limit of the matrix power M^k as k-->inf. results in a single column vector equal to the Fibonacci sequence. (End) Apparently a(n) = A128588(n-2) for n > 3. - Georg Fischer, Oct 14 2018 LINKS R. H. Hardin, Table of n, a(n) for n=1..100 FORMULA Empirical: a(n) = a(n-1) + a(n-2) for n >= 5. G.f.: (1-x^3)/(1-x-x^2) (conjecture). [Paul Barry, Aug 10 2009] a(n) = round(phi^(k-1)) - round(phi^(k-1)/sqrt(5)), phi = (1 + sqrt(5))/2 (conjecture). [Federico Provvedi, Mar 26 2013] G.f.: 1 + 2*x - x*Q(0), where Q(k) = 1 + x^2 - (2*k+1)*x + x*(2*k-1 - x)/Q(k+1);(conjecture), (continued fraction). - Sergei N. Gladkovskii, Oct 05 2013 G.f.: If prefaced with a 1, (1, 1, 1, 2, 2, 4,...): (1 - x^2 - x^4)/(1 - x - x^2); where the modified sequence satisfies A(x)/A(x^2), A(x) is the Fibonacci sequence. - Gary W. Adamson, Sep 15 2016 EXAMPLE All solutions for n=8 ...1.1...1.1...1.1...1.1...1.1...1.1...1.1...1.1...1.1...1.1...1.1...1.1...1.1 ...0.1...1.0...1.0...1.0...1.0...1.0...0.1...0.1...0.1...0.1...0.1...0.1...0.1 ...0.1...1.0...1.0...1.0...1.1...1.0...0.1...0.1...1.1...0.1...1.1...1.1...0.1 ...0.1...1.0...1.0...1.1...0.1...1.0...0.1...0.1...1.0...1.1...1.0...1.0...1.1 ...0.1...1.0...1.1...0.1...0.1...1.0...0.1...1.1...1.0...1.0...1.1...1.0...1.0 ...0.1...1.0...0.1...0.1...0.1...1.1...1.1...1.0...1.0...1.0...0.1...1.1...1.1 ...0.1...1.0...0.1...0.1...0.1...0.1...1.0...1.0...1.0...1.0...0.1...0.1...0.1 ...1.1...1.1...1.1...1.1...1.1...1.1...1.1...1.1...1.1...1.1...1.1...1.1...1.1 ------ ...1.1...1.1...1.1 ...1.0...1.0...1.0 ...1.0...1.1...1.1 ...1.1...0.1...0.1 ...0.1...0.1...1.1 ...1.1...1.1...1.0 ...1.0...1.0...1.0 ...1.1...1.1...1.1 MATHEMATICA Join[{1, 1}, Table[2*Fibonacci[n], {n, 70}]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *) Table[Round[GoldenRatio^(k-1)] - Round[GoldenRatio^(k-1)/Sqrt[5]], {k, 1, 70}] (* Federico Provvedi, Mar 26 2013 *) CROSSREFS Cf. A118658, A055389, A006355, A006355, A169985, A000045. Cf. A128588. - Georg Fischer, Oct 14 2018 Sequence in context: A006355 A055389 * A198834 A270925 A084202 A300865 Adjacent sequences:  A163730 A163731 A163732 * A163734 A163735 A163736 KEYWORD nonn,changed AUTHOR R. H. Hardin, Aug 03 2009 STATUS approved

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Last modified October 23 03:21 EDT 2018. Contains 316519 sequences. (Running on oeis4.)