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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)

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.

Sequence in context: A006355 A055389 * A198834 A270925 A084202 A053637

Adjacent sequences:  A163730 A163731 A163732 * A163734 A163735 A163736

KEYWORD

nonn

AUTHOR

R. H. Hardin, Aug 03 2009

STATUS

approved

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Last modified May 28 20:12 EDT 2017. Contains 287241 sequences.