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A320314 a(n) is the number of symmetric domino towers with n bricks.

%I A320314 #24 Aug 12 2022 09:23:04
%S A320314 1,1,3,3,7,9,19,25,53,71,149,203,423,583,1209,1681,3473,4863,10017,
%T A320314 14107,28987,41019,84113,119513,244645,348829,712987,1019731,2081547,
%U A320314 2985097,6086375,8749185,17820657,25671983,52241825,75402907,153316715,221673707,450393329,652234089
%N A320314 a(n) is the number of symmetric domino towers with n bricks.
%C A320314 A domino tower is a stack of bricks, where (1) each row is offset from the preceding row by half of a brick, (2) the bottom row is contiguous, and (3) each brick is supported from below by at least half of a brick.
%C A320314 The number of (not necessarily symmetric) domino towers with n blocks is given by 3^(n-1).
%C A320314 a(n) is odd for all n.
%C A320314 The not necessarily symmetric case is described in the Miklos Bona reference. Similar considerations lead to a decomposition of symmetric towers into half pyramids which are enumerated by the Motzkin numbers. - _Andrew Howroyd_, Mar 12 2021
%D A320314 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 25-27.
%H A320314 Andrew Howroyd, <a href="/A320314/b320314.txt">Table of n, a(n) for n = 1..1000</a>
%H A320314 Peter Kagey, <a href="https://math.stackexchange.com/q/2949131/121988">Symmetric Brick Stacking</a>, Mathematics Stack Exchange.
%F A320314 G.f.: (x + 2*x^3*M(x^2) + x^2*M(x^2))/((1-x^3*M(x^2))*(1-x^2*M(x^2))) where M(x) is the g.f. of A001006. - _Andrew Howroyd_, Mar 12 2021
%e A320314 For n = 4, the a(4) = 3 symmetric stacks are
%e A320314     +-------+
%e A320314     |       |
%e A320314 +---+---+---+---+
%e A320314 |       |       |
%e A320314 +---+---+---+---+,
%e A320314     |       |
%e A320314     +-------+
%e A320314 +-------+       +-------+
%e A320314 |       |       |       |
%e A320314 +---+---+---+---+---+---+, and
%e A320314     |       |       |
%e A320314     +-------+-------+
%e A320314 +-------+-------+-------+-------+
%e A320314 |       |       |       |       |
%e A320314 +-------+-------+-------+-------+.
%o A320314 (PARI) seq(n)={my(h=(1 - x^2 - sqrt(1-2*x^2-3*x^4 + O(x^3*x^n)))/(2*x^2)); Vec((x + 2*x*h + h)/((1-x*h)*(1-h)))} \\ _Andrew Howroyd_, Mar 12 2021
%Y A320314 Cf. A000244, A001006, A168368, A264746.
%K A320314 nonn
%O A320314 1,3
%A A320314 _Peter Kagey_, Oct 10 2018
%E A320314 a(20)-a(40) from _Andrew Howroyd_, Oct 25 2018