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

0,3

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

Links

Table of n, a(n) for n=0..34.

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

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

Example

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).

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 *)

Crossrefs

Cf. A000931, A108520.

Sequence in context: A244640 A230874 A349452 * A154919 A019230 A298148

Adjacent sequences: A176736 A176737 A176738 * A176740 A176741 A176742

Keyword

nonn,easy

Author

Wolfdieter Lang, Jul 14 2010

Status

approved