

A103770


A weighted tribonacci sequence, (1,3,9).


1



1, 1, 4, 16, 37, 121, 376, 1072, 3289, 9889, 29404, 88672, 265885, 796537, 2392240, 7174816, 21520369, 64574977, 193709428, 581117680, 1743420757, 5230158649, 15690480040, 47071742800, 141214610761, 423644159521, 1270933677004
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OFFSET

0,3


COMMENTS

The weighted tribonacci (1,r,r^2) with g.f. 1/(1xr*x^2r^2x^3) has general term sum{k=0..n, T(nk,k)r^k}.
Correspondence: a(n)=b(n+2)*3^n, where b(n) is the sequence of the arithmetic means of the previous three terms defined by b(n)=1/3*(b(n1)+b(n2)+b(n3)) with initial values b(0)=0, b(1)=0, b(2)=1; The g.f. for b(n) is B(x):=x^2/(1(x^1+x^2+x^3)/3), so the g.f. A(x) for a(n) suffices A(x)=B(3*x)/(3*x)^2. Because b(n) converges to the limit lim (1x)*B(x)=1/6*(b(0)+2*b(1)+3*b(2))=1/2 (for x>1), it follows that a(n)/3^n also converges to 1/2. This correspondence is valid in general (with necessary changes) for weighted sequences of order (1,p,p^2,p^3,p^4,...,p^(p1)) with natural p>0. Forming such sequences c(n):=c(n1)+p^1*c(n2)+...+p^(p1)*c(np) the limit of c(n)/p^n is 2/(p+1) (see also A001045).  Hieronymus Fischer, Feb 04 2006
a(n)/3^n equals the probability that n will occur as a partial sum in a randomlygenerated infinite sequence of 1s, 2s and 3s. The limit ratio is 1/2.  Bob Selcoe, Jul 05 2013
Number of compositions of n into one sort of 1's, three sorts of 2's, and nine sorts of 3's. [Joerg Arndt, Jul 06 2013]


LINKS

Table of n, a(n) for n=0..26.
Index to sequences with linear recurrences with constant coefficients, signature (1,3,9).


FORMULA

G.f.: 1/(1x3*x^29*x^3).
a(n) = sum{k=0..n, T(nk, k)3^k}, T(n, k) = trinomial coefficients (A027907).
a(n)=sum{k=0..n, sum{i=0..floor((nk)/2), C(nki, i)C(k, nki)}*3^(nk)};  Paul Barry, Apr 26 2005
a(n)/3^n converges to 1/2.  Hieronymus Fischer, Feb 02 2006
a(0)=1, a(1)=1, a(2)=4, a(n)=a(n1)+3*a(n2)+9*a(n3), n>=3;  Hieronymus Fischer, Feb 04 2006
a(n) = 3^n + b(n) + b(n1), with b(n) = (1)^A121262(n+1)*A088137(n+1).  Ralf Stephan, May 20 2007


CROSSREFS

Cf. A000073, A102001.
Cf. A071675.
Sequence in context: A080855 A203299 A198015 * A121318 A152133 A210440
Adjacent sequences: A103767 A103768 A103769 * A103771 A103772 A103773


KEYWORD

easy,nonn


AUTHOR

Paul Barry, Feb 15 2005


STATUS

approved



