OFFSET
1,2
COMMENTS
A problem important for polymer science because it counts the trees having unbranched branches; they are called "combs".
Equals (1, 3, 9, 27, 81, ...) convolved with (1, 0, 3, 9, 27, 81, ...). Example: a(5) = 162 = (81, 27, 9, 3, 1) dot (1, 0, 3, 9, 27) = 81 + 3*27. - Gary W. Adamson, Jul 31 2010
Floretion Algebra Multiplication Program, FAMP Code: lesforseq[ - 'i + 'j - 'kk' - 'ki' - 'kj' ], vesforseq(n) = 3^n, tesforseq = A006234
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (6,-9).
FORMULA
a(n) = 3*a(n-1) + 3^(n-2).
a(n) = (n+1)*3^(n-2), for n > 1.
From Paul Barry, Sep 05 2003: (Start)
a(n) = (n+2)3^(n-1) + 0^n/3 (offset 0).
A006234(n+4) - a(n+2) = 3^n. - Creighton Dement, Mar 01 2005
a(n+1) = Sum_{k=0..n} A196389(n,k)*3^k. - Philippe Deléham, Oct 31 2011
G.f.: (1 - 3*x + 3*x^2)*x/(1 - 3*x)^2. - Philippe Deléham, Oct 31 2011
a(n) = 6*a(n-1) - 9*a(n-2), with a(1)=1, a(2)=3, a(3)=12. - Harvey P. Dale, Feb 07 2012
E.g.f.: (exp(3*x)*(1 + 3*x) - 1)/9. - Stefano Spezia, Mar 05 2020
From Amiram Eldar, Jan 18 2021: (Start)
Sum_{n>=1} 1/a(n) = 27*log(3/2) - 19/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 17/2 - 27*log(4/3). (End)
EXAMPLE
a(5) = 162 because we can write (5+1)*3^(5-2) = 6*3^3 = 6*27.
MAPLE
a:=n->ceil(sum(3^(n-2), j=0..n)): seq(a(n), n=1..26); # Zerinvary Lajos, Jun 05 2008
MATHEMATICA
Join[{1}, Table[(n+1)3^(n-2), {n, 2, 30}]] (* or *) Join[{1}, LinearRecurrence[ {6, -9}, {3, 12}, 30]] (* Harvey P. Dale, Feb 07 2012 *)
PROG
(PARI) { for (n=1, 200, if (n>1, a=(n + 1)*p; p*=3, a=p=1); write("b064017.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 06 2009
(PARI) a(n)=if(n==1, 1, (n+1)*3^(n-2)); \\ Joerg Arndt, May 06 2013
(SageMath)
@CachedFunction
def BB(n, k, x): # modified cardinal B-splines
if n == 1: return 0 if (x < 0) or (x >= k) else 1
return x*BB(n-1, k, x) + (n*k-x)*BB(n-1, k, x-k)
def EulerianPolynomial(n, k, x):
if n == 0: return 1
return add(BB(n+1, k, k*m+1)*x^m for m in (0..n))
def A064017(n) : return 3^(n-1)*EulerianPolynomial(1, n-1, 1/3) if n != 1 else 1
[A064017(n) for n in (1..25)] # Peter Luschny, May 04 2013
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
Danail Bonchev (bonchevd(AT)aol.com), Sep 07 2001
STATUS
approved