A163714 -id:A163714 - cp's OEIS frontend

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.

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

Offset: 1

R. H. Hardin, Aug 03 2009

nonn

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->infinity 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

			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
   0 1   1 0   1 0   1 0   1 0   1 0   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
   0 1   1 0   1 0   1 1   0 1   1 0   0 1   0 1   1 0   1 1
   0 1   1 0   1 1   0 1   0 1   1 0   0 1   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 0   0 1   0 1   0 1   0 1   1 0   1 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
   0 1   0 1   0 1   1 0   1 0   1 0
   1 1   1 1   0 1   1 0   1 1   1 1
   1 0   1 0   1 1   1 1   0 1   0 1
   1 1   1 0   1 0   0 1   0 1   1 1
   0 1   1 1   1 1   1 1   1 1   1 0
   0 1   0 1   0 1   1 0   1 0   1 0
   1 1   1 1   1 1   1 1   1 1   1 1

R. H. Hardin, Table of n, a(n) for n=1..100

Cf. A118658, A055389, A006355, A006355, A169985, A000045.

Cf. A128588. - Georg Fischer, Oct 14 2018

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 *)

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

A163695 Number of n X 2 binary arrays with all 1s connected, a path of 1s from top row to lower right corner, and no 1 having more than two 1s adjacent.

Original entry on oeis.org

2, 5, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196, 12752043, 20633239, 33385282, 54018521, 87403803

Offset: 1

R. H. Hardin, Aug 03 2009

			All solutions for n=4:
...0.1...0.1...1.1...1.1...1.0...1.1...1.0...1.1...1.0...1.0...0.1
...0.1...0.1...0.1...0.1...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...1.0...1.0...0.1...0.1...1.0
...0.1...1.1...0.1...1.1...0.1...0.1...1.1...1.1...0.1...1.1...1.1

R. H. Hardin, Table of n, a(n) for n=1..100
Index entries for linear recurrences with constant coefficients, signature (1,1).

It appears that A163714 and A163733 have the same recurrence as this sequence.

Cf. A288219.

Vec(x*(2 - x)*(1 + x)^2 / (1 - x - x^2) + O(x^60)) \\ Colin Barker, Feb 20 2018

a(n) = a(n-1) + a(n-2) for n>=5.
[The Transfer Matrix Method provides this recurrence. - R. J. Mathar, Aug 02 2017]
From Colin Barker, Feb 20 2018: (Start)
G.f.: x*(2 - x)*(1 + x)^2 / (1 - x - x^2).
a(n) = (2^(-1-n)*((1-sqrt(5))^n*(-5+sqrt(5)) + (1+sqrt(5))^n*(5+sqrt(5)))) / sqrt(5) for n>2.
(End)

cp's OEIS Frontend

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.

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