A138978 Number of 3 X n matrices containing a 1 in the top left entry, all entries are integer values and adjacent entries differ by at most 1.

9, 121, 1665, 22979, 317259, 4380445, 60481881, 835088891, 11530288395, 159201677509, 2198138788809, 30350271502115, 419054058355851, 5785987905016141, 79888633386248025, 1103043049708026539, 15230001039404897259, 210284568423392013685, 2903453493049800669321

Offset: 1

Author

Wayne VanWeerthuizen, Apr 05 2008

Keywords

• nonn
• easy

Comments

Horizontally or vertically adjacent entries can differ by at most 1. Diagonally adjacent entries thus differ by at most 2.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..800
• Index entries for linear recurrences with constant coefficients, signature (16,-31,10).

Crossrefs

Cf. A138977, A138979.

Programs

Maple

a:= n-> (Matrix([1,6,2]). Matrix([[3,12,2], [2,10,2], [1,6,3]])^(n-1) .Matrix([[1],[1],[1]]))[1,1]: seq(a(n), n=1..20); # Alois P. Heinz, Aug 28 2008

Mathematica

LinearRecurrence[{16, -31, 10}, {9, 121, 1665}, 25] (* Paolo Xausa, Mar 17 2024 *)

Formula

a(n) = b(n)+c(n)+d(n), where b(1)=1, c(1)=6, d(1)=2, with b(n+1)=3*b(n)+2*c(n)+1*d(n), c(n+1)=12*b(n)+10*c(n)+6*d(n), d(n+1)=2*b(n)+2*c(n)+3*d(n).

G.f.: -x*(8*x^2-23*x+9) / (10*x^3-31*x^2+16*x-1). - Colin Barker, Dec 03 2012

Extensions

More terms from Alois P. Heinz, Aug 28 2008