A052909 Expansion of g.f. (1+x-x^2)/((1-x)*(1-3*x)).

1, 5, 16, 49, 148, 445, 1336, 4009, 12028, 36085, 108256, 324769, 974308, 2922925, 8768776, 26306329, 78918988, 236756965, 710270896, 2130812689, 6392438068, 19177314205, 57531942616, 172595827849, 517787483548, 1553362450645

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 889

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Formula

a(n) = 3*a(n-1) + 1, with a(0)=1, a(1)=5, a(2)=16.

a(n) = (11*3^n - 3)/6.

a(n) = 4*a(n-1) - 3*a(n-2). - Vincenzo Librandi, Jun 22 2012

a(n+1) = A237930(n) + 2*A000244(n). - Philippe Deléham, Feb 17 2014

a(n) = Sum_{k=1..3} floor((3^n)/k). - Lechoslaw Ratajczak, Jul 31 2016

E.g.f.: (11*exp(3*x) - 3*exp(x) - 2)/6. - Stefano Spezia, Aug 28 2023

Example

Ternary.......................Decimal
1...................................1
12..................................5
121................................16
1211...............................49
12111.............................148
121111............................445
1211111..........................1336
12111111.........................4009
121111111.......................12028
1211111111......................36085, etc. - Philippe Deléham, Feb 17 2014

Maple

spec := [S, {S=Prod(Union(Sequence(Z), Z), Sequence(Union(Z, Z, Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

Mathematica

CoefficientList[Series[(1+x-x^2)/((1-x)*(1-3*x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 22 2012 *)
Join[{1}, (11*3^Range[30] -3)/6] (* G. C. Greubel, Oct 15 2019 *)

Prog

(Magma) I:=[1, 5, 16]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012
(PARI) vector(30, n, if(n==1, 1, (11*3^(n-1) - 3)/6)) \\ G. C. Greubel, Oct 15 2019
(Sage) [1]+[(11*3^n -3)/6 for n in (1..30)] # G. C. Greubel, Oct 15 2019
(GAP) Concatenation([1], List([1..30], n-> (11*3^n - 3)/6)); # G. C. Greubel, Oct 15 2019

Crossrefs

Cf. A000244, A237930.

Sequence in context: A171426 A180129 A244410 * A037536 A192904 A082001

Adjacent sequences: A052906 A052907 A052908 * A052910 A052911 A052912

Keyword

nonn,easy

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Extensions

More terms from James A. Sellers, Jun 08 2000

Status

approved