A290604 a(0) = 2, a(1) = 2; for n > 1, a(n) = a(n-1) + 2*a(n-2) + 3.

2, 2, 9, 16, 37, 72, 149, 296, 597, 1192, 2389, 4776, 9557, 19112, 38229, 76456, 152917, 305832, 611669, 1223336, 2446677, 4893352, 9786709, 19573416, 39146837, 78293672, 156587349, 313174696, 626349397, 1252698792, 2505397589, 5010795176, 10021590357

Offset: 0

Author

Iain Fox, Oct 11 2017

Keywords

• easy
• nonn

Comments

Ratio of successive terms approaches 2.

Examples

			a(0) = 2.
a(1) = 2.
a(2) = 2 + 2*2 + 3 = 9.
a(3) = 9 + 2*2 + 3 = 16.
a(4) = 16 + 9*2 + 3 = 37.
...
		

Links

• Iain Fox, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2)

Programs

Magma

[(2^(n+2)+2*(-1)^n)/3+2^n-(3-(-1)^n)/2: n in [0..40]]; // Vincenzo Librandi, Oct 20 2017

Mathematica

Table[(2^(n + 2) + 2 (-1)^n) / 3 + 2^n - (3 - (-1)^n) / 2, {n, 0, 40}] (* Vincenzo Librandi, Oct 20 2017 *)
LinearRecurrence[{2,1,-2},{2,2,9},50] (* Harvey P. Dale, Mar 06 2025 *)

PARI

Vec((2/(1-x-2*x^2)) + (3*x^2/((1-x)*(1-x-2*x^2))) + O(x^50)) \\ Michel Marcus, Oct 12 2017

PARI

first(n) = Vec((2 - 2*x + 3*x^2)/(1 - 2*x - x^2 + 2*x^3) + O(x^n)) \\ Iain Fox, Dec 18 2017

Formula

a(n) = (2^(n+2) + 2*(-1)^n)/3 + 2^n - (3-(-1)^n)/2.

a(n) = A014113(n+1) + A141023(n).

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

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n > 2. - Iain Fox, Dec 18 2017