|
| |
|
|
A028859
|
|
a(n+2) = 2*a(n+1) + 2*a(n); a(0) = 1, a(1) = 3.
|
|
39
|
|
|
|
1, 3, 8, 22, 60, 164, 448, 1224, 3344, 9136, 24960, 68192, 186304, 508992, 1390592, 3799168, 10379520, 28357376, 77473792, 211662336, 578272256, 1579869184, 4316282880, 11792304128, 32217174016, 88018956288, 240472260608, 656982433792, 1794909388800, 4903783645184, 13397386067968
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,2
|
|
|
COMMENTS
|
Number of words of length n without adjacent 0's from the alphabet {0,1,2}. For example, a(2) counts 01,02,10,11,12,20,21,22. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 12 2001
Individually, both this sequence and A002605 are convergents to 1+sqrt(3). Mutually, both sequences are convergents to 2+sqrt(3) and 1+sqrt(3)/2. - Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Nov 04 2001 [Can someone clarify what is meant by the obscure second phrase, "Mutually..."? - M. F. Hasler, Aug 06 2018]
Add a loop at two vertices of the graph C_3=K_3. a(n) counts walks of length n+1 between these vertices. - Paul Barry, Oct 15 2004
Prefaced with a 1 as (1 + x + 3x^2 + 8x^3 + 22x^4 + ...) = 1 / (1 - x - 2x^2 - 3x^3 - 5x^4 - 8x^5 - 13x^6 - 21x^7 - ...). - Gary W. Adamson, Jul 28 2009
Pisano period lengths: 1, 1, 3, 1, 24, 3, 48, 1, 9, 24, 10, 3, 12, 48, 24, 1, 144, 9, 180, 24, .... - R. J. Mathar, Aug 10 2012
Also the number of independent vertex sets and vertex covers in the n-centipede graph. - Eric W. Weisstein, Sep 21 2017
Conjecture: Also the number of length n + 1 sequences that cover an initial interval of positive integers and whose non-adjacent parts are weakly decreasing. For example, (3,2,3,1,2) has non-adjacent pairs (3,3), (3,1), (3,2), (2,1), (2,2), (3,2), all of which are weakly decreasing, so is counted under a(11). The a(1) = 1 through a(3) = 8 sequences are:
(1) (11) (111)
(12) (121)
(21) (211)
(212)
(221)
(231)
(312)
(321)
The case of compositions is A333148, or A333150 for strict compositions, or A333193 for strictly decreasing parts. A version for ordered set partitions is A332872. Standard composition numbers of these compositions are A334966. Unimodal normal sequences are A227038. See also: A001045, A001523, A032020, A100471, A100881, A115981, A329398, A332836, A332872.
(End)
Number of 2-compositions of n+1 restricted to parts 1 and 2 (and allowed zeros); see Hopkins & Ouvry reference. - Brian Hopkins, Aug 16 2020
The number of ternary strings of length n not containing 00. Complement of A186244. - R. J. Mathar, Feb 13 2022
|
|
|
REFERENCES
|
S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 73).
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: -(x+1)/(2*x^2+2*x-1).
a(n) = [(1+sqrt(3))^(n+2)-(1-sqrt(3))^(n+2)]/(4*sqrt(3)). - Emeric Deutsch, Feb 01 2005
If p[i]=fibonacci(i+1) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. - Milan Janjic, May 08 2010
E.g.f.: exp(x)*(cosh(sqrt(3)*x) + 2*sinh(sqrt(3)*x)/sqrt(3)). - Stefano Spezia, Mar 02 2024
|
|
|
MAPLE
|
a[0]:=1:a[1]:=3:for n from 2 to 24 do a[n]:=2*a[n-1]+2*a[n-2] od: seq(a[n], n=0..24); # Emeric Deutsch
|
|
|
MATHEMATICA
|
a[n_]:=(MatrixPower[{{1, 3}, {1, 1}}, n].{{2}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)
Table[2^(n - 1) Hypergeometric2F1[(1 - n)/2, -n/2, -n, -2], {n, 20}] (* Eric W. Weisstein, Jun 14 2017 *)
|
|
|
PROG
|
(Haskell)
a028859 n = a028859_list !! n
a028859_list =
1 : 3 : map (* 2) (zipWith (+) a028859_list (tail a028859_list))
|
|
|
CROSSREFS
|
Cf. A155020 (same sequence with term 1 prepended).
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
