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
Louis Marin, Counting Polyominoes in a Rectangle b X h, arXiv:2406.16413 [cs.DM], 2024. See p. 145.
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
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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved