A307469 a(n) = 2*a(n-1) + 6*a(n-2) for n >= 2, a(0) = 1, a(1) = 5.

1, 5, 16, 62, 220, 812, 2944, 10760, 39184, 142928, 520960, 1899488, 6924736, 25246400, 92041216, 335560832, 1223368960, 4460102912, 16260419584, 59281456640, 216125430784, 787939601408, 2872631787520, 10472901183488, 38181593092096, 139200593285120

Offset

0,2

Comments

a(n) is the number of words of length n over alphabet {1,2,3,4,5} such that no odd letter is followed by an odd letter.

Links

Table of n, a(n) for n=0..25.

Index entries for linear recurrences with constant coefficients, signature (2,6).

Formula

a(n) = (-(2/7)*sqrt(7)+1/2)*(1-sqrt(7))^n+((2/7)*sqrt(7)+1/2)*(1+sqrt(7))^n.

G.f.: (1+3*x)/(1-2*x-6*x^2).

a(n) = 3*A083099(n) + A083099(n+1). - R. J. Mathar, Jan 27 2020

Example

For n=2 the a(2)=16 solutions are: 12, 14, 21, 22, 23, 24, 25, 32, 34, 41, 42, 43, 44, 45, 52, 54.

Maple

aseq:=proc(n) option remember;
if n<0 then return "seq not defined for negative indices";
elif n=0 then return 1;
elif n=1 then return 5;
else 2*aseq(n-1)+6*aseq(n-2);
end if;
end proc:
seq(aseq(n), n=0..26);

Mathematica

a[0] = 1; a[1] = 5;
a[n_] := a[n] = 2*a[n - 1] + 6*a[n - 2];
Table[a[n], {n, 0, 26}]
LinearRecurrence[{2, 6}, {1, 5}, 30] (* Harvey P. Dale, Feb 20 2023 *)

Crossrefs

The same over alphabet {1,2,3} gives A001045(n+2).

Sequence in context: A180719 A343164 A300317 * A197549 A365907 A034532

Adjacent sequences: A307466 A307467 A307468 * A307470 A307471 A307472

Keyword

nonn,easy

Author

Armend Shabani, Apr 09 2019

Status

approved