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
Examples
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
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).
Programs
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GAP
Concatenation([1], List([1..30], n-> (11*3^n - 3)/6)); # G. C. Greubel, Oct 15 2019
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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
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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 *) -
PARI
vector(30, n, if(n==1, 1, (11*3^(n-1) - 3)/6)) \\ G. C. Greubel, Oct 15 2019
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Sage
[1]+[(11*3^n -3)/6 for n in (1..30)] # G. C. Greubel, Oct 15 2019
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) = 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
Extensions
More terms from James Sellers, Jun 08 2000