login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A001224 If F(n) is the n-th Fibonacci number, then a(2n)=(F(2n+1)+F(n+2))/2 and a(2n+1)=(F(2n+2)+F(n+1))/2.
(Formerly M0318 N0117)
6
1, 2, 2, 4, 5, 9, 12, 21, 30, 51, 76, 127, 195, 322, 504, 826, 1309, 2135, 3410, 5545, 8900, 14445, 23256, 37701, 60813, 98514, 159094, 257608, 416325, 673933, 1089648, 1763581, 2852242, 4615823, 7466468, 12082291, 19546175, 31628466 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Arises from problem of finding the number of inequivalent ways to pack a 2 X n rectangle with dominoes. The official solution is given in A060312. The present sequence gives the correct answer provided n !=2, when it gives 2 instead of 1. To put it another way, the present sequence gives the number of tilings of a 2 x n rectangle with dominoes when left-to-right mirror images are not regarded as distinct. - N. J. A. Sloane, Mar 30 2015

Slavik V. Jablan (jablans(AT)yahoo.com) observes that this is also number of generating rational knots and links. See reference.

Also the number of distinct binding configurations on an n-site one-dimensional linear lattice, where the molecules cannot touch each other. This number determines the order of recurrence for the partition function of binding to a two-dimensional n X m lattice.

REFERENCES

S. Golomb, Polyominoes, Princeton Univ. Press 1994.

Jablan S. and Sazdanovic R., LinKnot: Knot Theory by Computer, World Scientific Press, 2007.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=1..500

Ali Reza Ashrafi, Jernej Azarija, Khadijeh Fathalikhani, Sandi Klavžar, Marko Petkovšek, Vertex and edge orbits of Fibonacci and Lucas cubes, arXiv:1407.4962 [math.CO], 2014.

M. Assis, J. L. Jacobsen, I. Jensen, J.-M. Maillard and B. M. McCoy, Integrability vs non-integrability: Hard hexagons and hard squares compared, arXiv preprint 1406.5566, 2014

C. G. Bower, Transforms (2)

S. V. Jablan, Geometry of Links, XII Yugoslav Geometric Seminar (Novi Sad, 1998), Novi Sad J. Math. 29 (1999), no. 3, 121-139.

Y. Kong, General recurrence theory of ligand binding on a three-dimensional lattice, J. Chem. Phys. Vol. 111 (1999), pp. 4790-4799. (Table I)

W. E. Patten (proposer) and S. W. Golomb (solver), Problem E1470, "Covering a 2Xn rectangle with dominoes", Amer. Math. Monthly, 69 (1962), 61-62.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

N. J. A. Sloane, Annotated scan of Monthly problem E1470 with illustration of a(4)=4 (Page 1)

N. J. A. Sloane, Annotated scan of Monthly problem E1470 with illustration of a(4)=4 (Page 2)

Index entries for sequences related to dominoes

FORMULA

From Christian G. Bower, May 09 2000: (Start)

G.f.: (2-(x+x^2)^2)/(2*(1-x-x^2)) + (1+x+x^2)*(x^2+x^4)/(2*(1-x^2-x^4)).

"BIK" transform of x+x^2. (End)

If F(n) is the n-th Fibonacci number, then a(2n)=(F(2n+1)+F(n+2))/2 and a(2n+1)=(F(2n+2)+F(n+1))/2.

MAPLE

# Maple code for A060312 and A001224 from N. J. A. Sloane, Mar 30 2015

with(combinat); F:=fibonacci;

f:=proc(n) option remember;

if n=2 then 1 # change this to 2 to get A001224

elif (n mod 2) = 0 then (F(n+1)+F(n/2+2))/2;

else (F(n+1)+F((n+1)/2))/2; fi; end;

[seq(f(n), n=1..50)];

A001224:=-(-1-z+2*z**2+z**3+z**4+z**5)/(z**4+z**2-1)/(z**2+z-1); # conjectured by Simon Plouffe in his 1992 dissertation

a:= n-> (Matrix([[5, 4, 2, 2, 1, 1]]). Matrix(6, (i, j)-> if (i=j-1) then 1 elif j=1 then [1, 2, -1, 0, -1, -1][i] else 0 fi)^n)[1, 6]: seq (a(n), n=1..38); # Alois P. Heinz, Aug 26 2008

MATHEMATICA

a[n_?EvenQ] := (Fibonacci[n + 1] + Fibonacci[n/2 + 2])/2; a[n_?OddQ] := (Fibonacci[n + 1] + Fibonacci[(n + 1)/2])/2; Table[a[n], {n, 38}] (* Jean-François Alcover, Oct 06 2011, after formula *)

LinearRecurrence[{1, 2, -1, 0, -1, -1}, {1, 2, 2, 4, 5, 9}, 38] (* Jean-François Alcover, Sep 21 2017 *)

PROG

(MAGMA) [(1/2)*((Fibonacci(n+1))+Fibonacci(Floor((n+3+(-1)^n) div 2))): n in [1..40]]; // Vincenzo Librandi, Nov 23 2014

CROSSREFS

Essentially the same as A060312, A068928 and A102526.

Sequence in context: A124280 A088518 * A102526 A050192 A191786 A007147

Adjacent sequences:  A001221 A001222 A001223 * A001225 A001226 A001227

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Christian G. Bower, May 09 2000

Typo in references corrected by Jernej Azarija, Oct 23 2013

Edited by N. J. A. Sloane, Mar 30 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 12 16:18 EST 2017. Contains 295939 sequences.