

A034182


Number of notnecessarilysymmetric n X 2 crossword puzzle grids.


20



1, 5, 15, 39, 97, 237, 575, 1391, 3361, 8117, 19599, 47319, 114241, 275805, 665855, 1607519, 3880897, 9369317, 22619535, 54608391, 131836321, 318281037, 768398399, 1855077839, 4478554081, 10812186005, 26102926095, 63018038199, 152139002497, 367296043197
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OFFSET

1,2


COMMENTS

n X 2 binary arrays with a path of adjacent 1's and no path of adjacent 0's from top row to bottom row.  R. H. Hardin, Mar 21 2002
Define a triangle with T(n,1) = T(n,n) = n*(n1) + 1, n>=1, and its interior terms via T(r,c) = T(r1,c) + T(r1,c1)+ T(r2,c1), 2<=c<r. This gives 1; 3,3; 7,7,7; 13,17,17,13; 21,37,41,37,21; etc. The row sums are 1, 6, 21, 60, 157, 394, etc., and the first differences of the row sums are this sequence.  J. M. Bergot, Mar 16 2013


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,1)


FORMULA

a(n) = 2a(n1) + a(n2) + 4.
a(n) = 2 + (3/2)*(1+sqrt(2))^(n1)  sqrt(2)*(1sqrt(2))^(n1) + (3/2)*(1sqrt(2))^(n1) + (1+sqrt(2))^(n1)*sqrt(2), with n>=1  Paolo P. Lava, Jun 10 2008
(1 + 5x + 15x^2 + ...) = (1 + 2x + 2x^2 + ...) * (1 + 3x + 7x^2 + ...), where A001333 = (1, 1, 3, 7, 17, 41, ...).
a(n) = (4 + (1sqrt(2))^(1+n) + (1+sqrt(2))^(1+n))/2. G.f.: x*(1+x)^2/((1x)*(1  2*x  x^2)).  Colin Barker, May 22 2012
a(n) = A001333(n+1)2.  R. J. Mathar, Mar 28 2013


MATHEMATICA

t={0, 0}; Do[AppendTo[t, t[[2]]+2*t[[1]]+2], {n, 40}]; Drop[t1, 2] (* Vladimir Joseph Stephan Orlovsky, Jan 25 2012 *)
{1}~Join~NestList[{#2, 2 #2 + #1 + 4} & @@ # &, {1, 5}, 28][[All, 1]] (* Michael De Vlieger, Oct 02 2017 *)


PROG

(Haskell)
a034182 n = a034182_list !! (n1)
a034182_list = 1 : 5 : (map (+ 4) $
zipWith (+) a034182_list (map (* 2) $ tail a034182_list))
 Reinhard Zumkeller, May 23 2013


CROSSREFS

Row 2 of A292357.
Column sums of A059678.
Cf. A001333, A034184, A034187.
Sequence in context: A084447 A099035 A262295 * A132985 A022570 A152881
Adjacent sequences: A034179 A034180 A034181 * A034183 A034184 A034185


KEYWORD

nonn,easy


AUTHOR

Erich Friedman


STATUS

approved



