A034182 Number of not-necessarily-symmetric n X 2 crossword puzzle grids.

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

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*(n-1) + 1, n>=1, and its interior terms via T(r,c) = T(r-1,c) + T(r-1,c-1)+ T(r-2,c-1), 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(n-1) + a(n-2) + 4.

(1 + 5x + 15x^2 + ...) = (1 + 2x + 2x^2 + ...) * (1 + 3x + 7x^2 + ...), convolution of A040000 and left-shifted A001333.

a(n) = (-4 + (1-sqrt(2))^(1+n) + (1+sqrt(2))^(1+n))/2. G.f.: x*(1+x)^2/((1-x)*(1 - 2*x - x^2)). - Colin Barker, May 22 2012

a(n) = A001333(n+1)-2. - R. J. Mathar, Mar 28 2013

a(n) = A048739(n-3) +2*A048739(n-2) +A048739(n-1). - R. J. Mathar, Jun 15 2020

Mathematica

{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 !! (n-1)
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, A052542 (first differences).

Sequence in context: A357290 A099035 A262295 * A348885 A132985 A022570

Adjacent sequences: A034179 A034180 A034181 * A034183 A034184 A034185

Keyword

nonn,easy

Author

Erich Friedman

Status

approved