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%I #40 Jul 15 2022 18:48:20
%S 1,3,7,15,33,73,161,355,783,1727,3809,8401,18529,40867,90135,198799,
%T 438465,967065,2132929,4704323,10375711,22884351,50473025,111321761,
%U 245527873,541528771,1194379303,2634286479,5810101729,12814582761
%N 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 Column 1 of A193648.
%C Or yet empirical: row sums of triangle
%C m/k | 0 1 2 3 4 5 6 7
%C ==================================================
%C 0 | 1
%C 1 | 1 2
%C 2 | 1 2 4
%C 3 | 1 2 4 8
%C 4 | 1 4 4 8 16
%C 5 | 1 4 12 8 16 32
%C 6 | 1 4 12 32 16 32 64
%C 7 | 1 6 12 32 80 32 64 128
%C which is triangle for numbers 2^k*C(m,k) with triplicated diagonals. - _Vladimir Shevelev_, Apr 13 2012
%H R. H. Hardin, <a href="/A193641/b193641.txt">Table of n, a(n) for n = 1..200</a>
%H Tomislav Doslic and I. 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 Empirical: a(n) = 2*a(n-1) + a(n-3).
%F 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 Empirical: a(n) = 1 + 2*A077852(n-2) for n >= 2. - _Greg Dresden_, Apr 04 2021
%F Empirical: partial sums of A052910. - _Sean A. Irvine_, Jul 14 2022
%e Some solutions for n=6:
%e 1 1 1 0 0 1 -1 1 0 -1 -1 0 0 0 -1 -1
%e -1 -1 -1 0 -1 -1 1 -1 1 1 1 1 1 0 1 1
%e -1 0 1 0 1 1 0 0 -1 -1 0 -1 -1 1 -1 1
%e 1 1 1 0 1 0 -1 -1 1 1 0 0 -1 -1 -1 -1
%e 0 -1 -1 -1 -1 0 1 1 -1 0 0 0 1 1 1 1
%e 0 1 1 1 1 0 -1 0 0 0 0 0 0 -1 -1 -1
%o (Haskell)
%o a193641 n = a193641_list !! n
%o a193641_list = drop 2 xs where
%o xs = 1 : 1 : 1 : zipWith (+) xs (map (* 2) $ drop 2 xs)
%o -- _Reinhard Zumkeller_, Jan 01 2014
%K nonn
%O 1,2
%A _R. H. Hardin_, Aug 02 2011
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