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A034182 Number of not-necessarily-symmetric n X 2 crossword puzzle grids. 20

%I #44 Dec 30 2023 23:46:22

%S 1,5,15,39,97,237,575,1391,3361,8117,19599,47319,114241,275805,665855,

%T 1607519,3880897,9369317,22619535,54608391,131836321,318281037,

%U 768398399,1855077839,4478554081,10812186005,26102926095,63018038199,152139002497,367296043197

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

%C 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

%C 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

%H Reinhard Zumkeller, <a href="/A034182/b034182.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-1)

%F a(n) = 2a(n-1) + a(n-2) + 4.

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

%F 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

%F a(n) = A001333(n+1)-2. - _R. J. Mathar_, Mar 28 2013

%F a(n) = A048739(n-3) +2*A048739(n-2) +A048739(n-1). - _R. J. Mathar_, Jun 15 2020

%t {1}~Join~NestList[{#2, 2 #2 + #1 + 4} & @@ # &, {1, 5}, 28][[All, -1]] (* _Michael De Vlieger_, Oct 02 2017 *)

%o (Haskell)

%o a034182 n = a034182_list !! (n-1)

%o a034182_list = 1 : 5 : (map (+ 4) $

%o zipWith (+) a034182_list (map (* 2) $ tail a034182_list))

%o -- _Reinhard Zumkeller_, May 23 2013

%Y Row 2 of A292357.

%Y Column sums of A059678.

%Y Cf. A001333, A034184, A034187, A052542 (first differences).

%K nonn,easy

%O 1,2

%A _Erich Friedman_

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