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

1,2

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

Links

R. H. Hardin, Table of n, a(n) for n = 1..200

Tomislav Doslic and I. Zubac, Counting maximal matchings in linear polymers, Ars Mathematica Contemporanea 11 (2016) 255-276.

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

Example

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

Prog

(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

Crossrefs

Sequence in context: A269560 A146654 A363503 * A026701 A249512 A140498

Adjacent sequences: A193638 A193639 A193640 * A193642 A193643 A193644

Keyword

nonn

Author

R. H. Hardin, Aug 02 2011

Status

approved