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.

1, 3, 7, 15, 33, 73, 161, 355, 783, 1727, 3809, 8401, 18529, 40867, 90135, 198799, 438465, 967065, 2132929, 4704323, 10375711, 22884351, 50473025, 111321761, 245527873, 541528771, 1194379303, 2634286479, 5810101729, 12814582761

Offset: 1

Author

R. H. Hardin, Aug 02 2011

Keywords

• nonn

Comments

Column 1 of A193648.

Or yet empirical: row sums of triangle

m/k | 0 1 2 3 4 5 6 7

==================================================

0 | 1

1 | 1 2

2 | 1 2 4

3 | 1 2 4 8

4 | 1 4 4 8 16

5 | 1 4 12 8 16 32

6 | 1 4 12 32 16 32 64

7 | 1 6 12 32 80 32 64 128

which is triangle for numbers 2^k*C(m,k) with triplicated diagonals. - Vladimir Shevelev, Apr 13 2012

Examples

			Some solutions for n=6:
   1   1   1   0   0   1  -1   1   0  -1  -1   0   0   0  -1  -1
  -1  -1  -1   0  -1  -1   1  -1   1   1   1   1   1   0   1   1
  -1   0   1   0   1   1   0   0  -1  -1   0  -1  -1   1  -1   1
   1   1   1   0   1   0  -1  -1   1   1   0   0  -1  -1  -1  -1
   0  -1  -1  -1  -1   0   1   1  -1   0   0   0   1   1   1   1
   0   1   1   1   1   0  -1   0   0   0   0   0   0  -1  -1  -1
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..200
• Jean-Luc Baril and Nathanaël Hassler, Intervals in a family of Fibonacci lattices, Univ. de Bourgogne (France, 2024). See p. 7.
• Tomislav Doslic and Ivana Zubac, Counting maximal matchings in linear polymers, Ars Mathematica Contemporanea 11 (2016) 255-276.

Programs

Haskell

a193641 n = a193641_list !! n
a193641_list = drop 2 xs where
   xs = 1 : 1 : 1 : zipWith (+) xs (map (* 2) $ drop 2 xs)
-- Reinhard Zumkeller, Jan 01 2014

Formula

Empirical: a(n) = 2*a(n-1) + a(n-3).

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

Empirical: a(n) = 1 + 2*A077852(n-2) for n >= 2. - Greg Dresden, Apr 04 2021

Empirical: partial sums of A052910. - Sean A. Irvine, Jul 14 2022