A068397 a(n) = Lucas(n) + (-1)^n + 1.

1, 5, 4, 9, 11, 20, 29, 49, 76, 125, 199, 324, 521, 845, 1364, 2209, 3571, 5780, 9349, 15129, 24476, 39605, 64079, 103684, 167761, 271445, 439204, 710649, 1149851, 1860500, 3010349, 4870849, 7881196, 12752045, 20633239, 33385284, 54018521, 87403805

Offset: 1

Author

Sharon Sela (sharonsela(AT)hotmail.com), Mar 30 2002

Keywords

• nonn
• easy

Comments

Number of domino tilings of a 2 X n strip on a cylinder.

Number of domino tilings of a 2 X n rectangle = Fibonacci(n) - see A000045.

Number of perfect matchings in the C_n X P_2 graph (C_n is the cycle graph on n vertices and P_2 is the path graph on 2 vertices). - Emeric Deutsch, Dec 29 2004

For n >= 3, also the number of maximum independent edge sets (matchings) in the n-prism graph. - Eric W. Weisstein, Mar 30 2017

For n >= 4, also the number of minimum clique coverings in the n-prism graph. - Eric W. Weisstein, May 03 2017

Examples

			G.f. = 5*x^2 + 4*x^3 + 9*x^4 + 11*x^5 + 20*x^6 + 29*x^7 + 49*x^8 + 76*x^9 + ...
Example: a(3)=4 because in the graph with vertex set {A,B,C,A',B',C'} and edge set {AB,AC,BC, A'B',A'C',B'C',AA',BB',CC'} we have the following perfect matchings: {AA',BC,B'C'}, {BB',AC,A'C'}, {CC',AB,A'B'} and {AA',BB',CC'}.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000 (corrected by Michel Marcus, Jan 19 2019)
• Cate S. Anstöter, Nino Bašić, Patrick W. Fowler, and Tomaž Pisanski, Catacondensed Chemical Hexagonal Complexes: A Natural Generalisation of Benzenoids, arXiv:2104.13290 [physics.chem-ph], 2021.
• M. Baake, J. Hermisson, and P. Pleasants, The torus parametrization of quasiperiodic LI-classes J. Phys. A 30 (1997), no. 9, 3029-3056. See Table 3.
• Sarah-Marie Belcastro, Domino Tilings of 2 X n Grids (or Perfect Matchings of Grid Graphs) on Surfaces, J. Integer Seq. 26 (2023), Article 23.5.6.
• H. Hosoya and F. Harary, On the matching properties of three fence graphs, J. Math. Chem., 12(1993), 211-218.
• H. Hosoya and A. Motoyama, An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices, J. Math. Physics 26 (1985) 157-167 (eq. (21) and Table IV).
• B. Myers, Number of spanning trees in a wheel, IEE Trans. Circuit Theo. 18 (2) (1971) 280-282, Table 1.
• Eric Weisstein's World of Mathematics, Clique Covering
• Eric Weisstein's World of Mathematics, Matching
• Eric Weisstein's World of Mathematics, Maximum Independent Edge Set
• Eric Weisstein's World of Mathematics, Minimum Edge Cover
• Eric Weisstein's World of Mathematics, Prism Graph
• Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1).

Crossrefs

Cf. A000032, A000045.

Cf. also A102079, A102091, A252054.

a(n) = A102079(n, n).

Programs

Maple

a[2]:=5: a[3]:=4: a[4]:=9: a[5]:=11: for n from 6 to 45 do a[n]:=a[n-1]+2*a[n-2]-a[n-3]-a[n-4] od:seq(a[n],n=2..40); # Emeric Deutsch, Dec 29 2004
f:= n -> combinat:-fibonacci(n-1)+combinat:-fibonacci(n+1)+(-1)^n+1:
map(f, [$1..50]); # Robert Israel, May 03 2017

Mathematica

Table[LucasL[n] + (-1)^n + 1, {n, 1, 38}] (* Jean-François Alcover, Sep 01 2011 *)
LucasL[#] + (-1)^# + 1 &[Range[38]] (* Eric W. Weisstein, May 03 2017 *)
LinearRecurrence[{1, 2, -1, -1}, {1, 5, 4, 9}, 20] (* Eric W. Weisstein, Dec 31 2017 *)
CoefficientList[Series[(1 + 4 x - 3 x^2 - 4 x^3)/(1 - x - 2 x^2 + x^3 + x^4), {x, 0, 20}], x]

PARI

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,-1,2,1]^(n-1)*[1;5;4;9])[1,1] \\ Charles R Greathouse IV, Jun 19 2016

PARI

Vec(x*(1+4*x-3*x^2-4*x^3)/(1-x-2*x^2+x^3+x^4) + O(x^40)) \\ Colin Barker, Jan 28 2017; Michel Marcus, Jan 19 2019

Formula

a(n) = F(n+1) + F(n-1) + 2 if n is even, a(n) = F(n+1) + F(n-1) if n is odd, where F(n) is the n-th Fibonacci number - sequence A000045.

a(n) = 1 + (-1)^n + ((1 + sqrt(5))/2)^n + ((1 - sqrt(5))/2)^n = 1 + (-1)^n + A000032(n). - Vladeta Jovovic, Apr 08 2002

Recurrence: a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4). - Vladeta Jovovic, Apr 08 2002

a(n+2) = a(n+1) + a(n) if n even, a(n+2) = a(n+1) + a(n) + 2 if n odd. - Michael Somos, Jan 28 2017

a(1) = 1, a(2) = 5; a(n) = a(n-1) + a(n-2) - 2*(n mod 2). [Belcastro]

G.f.: x*(1 + 4*x - 3*x^2 - 4*x^3)/(1 - x - 2*x^2 + x^3 + x^4). - Vladeta Jovovic, Apr 08 2002

a(n) = ((1 + sqrt(5))/2)^n + ((1 - sqrt(5))/2)^n + 1 + (-1)^n. [Hosoya/Harary]

E.g.f.: exp(-x/phi) + exp(phi*x) + 2*cosh(x) - 4, where phi is the golden ratio. - Ilya Gutkovskiy, Jun 16 2016

Extensions

More terms from Vladeta Jovovic, Apr 08 2002

Two initial terms added, third comment amended to be consonant with new initial terms, offset changed to be consonant with initial terms, two references added, two formulas added. - Sarah-Marie Belcastro, Jul 04 2009

Edited by N. J. A. Sloane, Jan 10 2018 to incorporate information from a duplicate (but now dead) entry.