A176739 Expansion of 1/(1-2*x^2-4*x^3). (2,4)-Padovan sequence.

1, 0, 2, 4, 4, 16, 24, 48, 112, 192, 416, 832, 1600, 3328, 6528, 13056, 26368, 52224, 104960, 209920, 418816, 839680, 1677312, 3354624, 6713344, 13418496, 26845184, 53690368, 107364352, 214761472, 429490176, 858980352, 1718026240, 3435921408, 6871973888

Offset: 0

Author

Wolfdieter Lang, Jul 14 2010

Keywords

• nonn
• easy

Comments

See A000931 (Padovan), and the W. Lang link given there for a combinatorial interpretation and an explicit form.

a(n)/2^n equals the probability that n will occur as a partial sum in a randomly-generated infinite sequence of 2's and 3's. The limiting ratio is 2/5. - Bob Selcoe, Jul 12 2013

Examples

			Combinatorics for (A,B)=(2,4) Padovan sequence with weighted (3,2)-Morse type code (see the W. Lang link under A000931): n=5, - -- and -- -, with weights 2^1*4^1 and 4^1*2^1, respectively, adding to 2*2*4=16=a(5).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0,2,4).

Crossrefs

Cf. A000931, A108520.

Programs

Maple

seq(2^(n+1)/5 + Re((3-I)*(-1-I)^n)/5, n=0..100); # Robert Israel, Aug 26 2014

Mathematica

CoefficientList[Series[1/(1 - 2*x^2 - 4*x^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 26 2014 *)

Formula

O.g.f.: 1/((1+2*x+2*x^2)*(1-2*x)) = ((3+2*x)/(1+2*x+2*x^2) + 2/(1-2*x))/5.

a(n) = (3*b(n) + 2*b(n-1) + 2^(n+1))/5, with b(n):=A108520(n), and b(-1)=0.

a(n) = 2*a(n-2) + 4*a(n-3). - Bob Selcoe, Aug 26 2014

a(n) = 2^(n+1)/5 + Re((3-i)*(-1-i)^n)/5. - Robert Israel, Aug 26 2014

5*a(n) = 2^(n+1) -A078069(n+1). - R. J. Mathar, May 14 2024