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A193641 Number of arrays of -1..1 integers x(1..n) with every x(i) in a subsequence of length 1 or 2 with sum zero.

%I A193641 #43 Dec 03 2024 11:16:35
%S A193641 1,3,7,15,33,73,161,355,783,1727,3809,8401,18529,40867,90135,198799,
%T A193641 438465,967065,2132929,4704323,10375711,22884351,50473025,111321761,
%U A193641 245527873,541528771,1194379303,2634286479,5810101729,12814582761
%N A193641 Number of arrays of -1..1 integers x(1..n) with every x(i) in a subsequence of length 1 or 2 with sum zero.
%C A193641 Column 1 of A193648.
%C A193641 Or yet empirical: row sums of triangle
%C A193641   m/k |  0     1     2     3     4     5     6     7
%C A193641   ==================================================
%C A193641    0  |  1
%C A193641    1  |  1     2
%C A193641    2  |  1     2     4
%C A193641    3  |  1     2     4     8
%C A193641    4  |  1     4     4     8    16
%C A193641    5  |  1     4    12     8    16    32
%C A193641    6  |  1     4    12    32    16    32    64
%C A193641    7  |  1     6    12    32    80    32    64   128
%C A193641 which is triangle for numbers 2^k*C(m,k) with triplicated diagonals. - _Vladimir Shevelev_, Apr 13 2012
%H A193641 R. H. Hardin, <a href="/A193641/b193641.txt">Table of n, a(n) for n = 1..200</a>
%H A193641 Jean-Luc Baril and Nathanaël Hassler, <a href="http://jl.baril.u-bourgogne.fr/interva.pdf">Intervals in a family of Fibonacci lattices</a>, Univ. de Bourgogne (France, 2024). See p. 7.
%H A193641 Tomislav Doslic and Ivana Zubac, <a href="https://doi.org/10.26493/1855-3974.851.167">Counting maximal matchings in linear polymers</a>, Ars Mathematica Contemporanea 11 (2016) 255-276.
%F A193641 Empirical: a(n) = 2*a(n-1) + a(n-3).
%F A193641 Empirical: G.f.: -x*(1+x+x^2) / ( -1+2*x+x^3 ); a(n) = A008998(n-3) + A008998(n-2) + A008998(n-1). - _R. J. Mathar_, Feb 19 2015
%F A193641 Empirical: a(n) = 1 + 2*A077852(n-2) for n >= 2. - _Greg Dresden_, Apr 04 2021
%F A193641 Empirical: partial sums of A052910. - _Sean A. Irvine_, Jul 14 2022
%e A193641 Some solutions for n=6:
%e A193641    1   1   1   0   0   1  -1   1   0  -1  -1   0   0   0  -1  -1
%e A193641   -1  -1  -1   0  -1  -1   1  -1   1   1   1   1   1   0   1   1
%e A193641   -1   0   1   0   1   1   0   0  -1  -1   0  -1  -1   1  -1   1
%e A193641    1   1   1   0   1   0  -1  -1   1   1   0   0  -1  -1  -1  -1
%e A193641    0  -1  -1  -1  -1   0   1   1  -1   0   0   0   1   1   1   1
%e A193641    0   1   1   1   1   0  -1   0   0   0   0   0   0  -1  -1  -1
%o A193641 (Haskell)
%o A193641 a193641 n = a193641_list !! n
%o A193641 a193641_list = drop 2 xs where
%o A193641    xs = 1 : 1 : 1 : zipWith (+) xs (map (* 2) $ drop 2 xs)
%o A193641 -- _Reinhard Zumkeller_, Jan 01 2014
%K A193641 nonn
%O A193641 1,2
%A A193641 _R. H. Hardin_, Aug 02 2011