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A183712 1/20 of the number of (n+1) X 3 0..4 arrays with every 2 X 2 subblock strictly increasing clockwise or counterclockwise with one decrease. 1
5, 17, 54, 174, 559, 1797, 5776, 18566, 59677, 191821, 616574, 1981866, 6370351, 20476345, 65817520, 211558554, 680016837, 2185791545, 7025832918, 22583273462, 72589861759, 233327025821, 749987665760, 2410700161342, 7748761123965, 24906995867477 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Column 2 of A183719. [Corrected by M. F. Hasler, Oct 07 2014]

This sequence counts closed walks of length (n+2) at the vertex of a triangle, to which a loop has been added to one of the remaining vertices and two loops has been added to the third vertex. - David Neil McGrath, Sep 04 2014

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..200

Index entries for linear recurrences with constant coefficients, signature (3,1,-1).

FORMULA

a(n) = 3*a(n-1) + a(n-2) - a(n-3).

The top left element of A^(n+2) where A=(0,1,1;1,1,1;1,1,2). - David Neil McGrath, Sep 04 2014

a(n) ~ c*k^n where k = 1.629316... is the largest root of x^3 - 3x^2 - x + 1 and c = 1.6293... is conjecturally the largest root of 148x^3 - 296x^2 + 90x - 1. - Charles R Greathouse IV, Sep 15 2014

G.f.: x*(5+2*x-2*x^2) / (1-3*x-x^2+x^3). - Colin Barker, Mar 16 2016

EXAMPLE

Some solutions for 5 X 3:

..0..1..4....1..2..0....4..0..4....4..3..4....4..0..4....1..4..0....3..4..2

..3..2..3....0..3..4....2..1..3....0..2..0....3..2..3....2..3..1....1..0..1

..4..1..0....1..2..1....4..0..4....4..3..4....0..1..0....0..4..0....2..4..3

..3..2..3....0..3..4....3..2..3....0..2..1....4..2..3....1..3..1....1..0..1

..4..0..4....1..2..1....4..1..0....4..3..0....0..1..0....0..4..0....2..3..2

...

...R..L.......R..L.......R..L.......L..R.......R..L.......L..R.......R..L...

...L..R.......L..R.......L..R.......R..L.......L..R.......R..L.......L..R...

...R..L.......R..L.......R..L.......L..R.......R..L.......L..R.......R..L...

...L..R.......L..R.......L..R.......R..L.......L..R.......R..L.......L..R...

PROG

(PARI) a(n)=([0, 1, 1; 1, 1, 1; 1, 1, 2]^(n+2))[1, 1] \\ Charles R Greathouse IV, Sep 15 2014

(PARI) Vec(x*(5+2*x-2*x^2)/(1-3*x-x^2+x^3) + O(x^50)) \\ Colin Barker, Mar 16 2016

CROSSREFS

Cf. A183727, A183719.

Sequence in context: A034346 A055419 A027091 * A244235 A081495 A191645

Adjacent sequences:  A183709 A183710 A183711 * A183713 A183714 A183715

KEYWORD

nonn,walk,easy

AUTHOR

R. H. Hardin, Jan 06 2011

STATUS

approved

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Last modified December 9 06:07 EST 2021. Contains 349627 sequences. (Running on oeis4.)