A051926 Number of independent sets of nodes in graph C_4 X P_n (n>2).

1, 7, 35, 181, 933, 4811, 24807, 127913, 659561, 3400911, 17536203, 90422365, 466247117, 2404121747, 12396433487, 63920042065, 329592522065, 1699486218903, 8763103574515, 45185411569413, 232990675202677, 1201375684008283, 6194683683674679, 31941803427179001

Offset: 0

Author

Stephen G Penrice, Dec 19 1999

Keywords

• easy
• nonn

Comments

Number of ways zero or more black and white stones can be placed on the points of a 2 X n grid such that no white stones are adjacent to any black stones. A078057 is a related case, where the grid is 1 X n. - Wayne VanWeerthuizen, May 04 2004

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8.
• Sela Fried and Toufik Mansour, Staircase graph words, arXiv:2312.08273 [math.CO], 2023.
• Index entries for linear recurrences with constant coefficients, signature (5,1,-1).

Crossrefs

Row 4 of A286513.

Programs

Magma

I:=[1, 7, 35]; [n le 3 select I[n] else 5*Self(n-1)+Self(n-2)-Self(n-3): n in [1..25]]; // Vincenzo Librandi, Apr 27 2012

Mathematica

CoefficientList[Series[(1+2*x-x^2)/(1-5*x-x^2+x^3),{x,0,30}],x] (* Vincenzo Librandi, Apr 27 2012 *)
LinearRecurrence[{5,1,-1},{1,7,35},40] (* Harvey P. Dale, Apr 29 2019 *)

Formula

a(n) = 5*a(n-1)+a(n-2)-a(n-3) for n>2. - Wayne VanWeerthuizen, May 04 2004

G.f.: (1+2*x-x^2)/(1-5*x-x^2+x^3). - Colin Barker, Apr 18 2012

Extensions

More terms from James Sellers, Dec 20 1999