A083424 a(n) = (5*4^n + (-2)^n)/6.

1, 3, 14, 52, 216, 848, 3424, 13632, 54656, 218368, 873984, 3494912, 13981696, 55922688, 223698944, 894779392, 3579150336, 14316535808, 57266274304, 229064835072, 916259864576, 3665038409728, 14660155736064, 58640618749952

Offset: 0

Author

Paul Barry, Apr 30 2003

Keywords

• easy
• nonn

Comments

Binomial transform of A083423.

Examples

			Factorizations of initial terms: 1, (3), (2)*(7), (2)^2*(13), (2)^3*(3)^3, (2)^4*(53), (2)^5*(107), (2)^6*(3)*(71), (2)^7*(7)*(61), (2)^8*(853), (2)^9*(3)*(569), (2)^10*(3413), (2)^11*(6827), (2)^12*(3)^2*(37)*(41), (2)^13*(7)*(47)*(83), (2)^14*(13)*(4201), (2)^15*(3)*(23)*(1583), (2)^16*(218453), ...
		

Links

• Index entries for linear recurrences with constant coefficients, signature (2,8).

Crossrefs

Cf. A000079, A048573, A083423.

Programs

Maple

A083424:=n->(5*4^n+(-2)^n)/6; [seq(A083424(n),n=0..50)]; # N. J. A. Sloane, Jul 18 2014

Mathematica

LinearRecurrence[{2,8},{1,3},30] (* Harvey P. Dale, Apr 21 2019 *)

PARI

a(n)=(5*4^n+(-2)^n)/6 \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = 2*a(n-1) + 8*a(n-2). - N. J. A. Sloane, Jul 16 2014

G.f.: (1+x)/(1-2*x-8*x^2). [Corrected by N. J. A. Sloane, Jul 16 2014]

E.g.f.: (5*exp(4*x) + exp(-2*x))/6.

From N. J. A. Sloane, Jul 18 2014: (Start)

2^(n-1)|a(n) for n >= 1;

3|a(3n+1). (End)

From Klaus Purath, Oct 15 2020: (Start)

a(n) = A048573(n)*2^(n-1).

a(n) = A048573(n)*(A048573(n+1) - A048573(n-1))/5. (End)

a(n+1) = 4*a(n) - (-1)^n*A000079(n). - Paul Curtz, May 22 2025