A064017 Number of ternary trees (A001764) with n nodes and maximal diameter.

1, 3, 12, 45, 162, 567, 1944, 6561, 21870, 72171, 236196, 767637, 2480058, 7971615, 25509168, 81310473, 258280326, 817887699, 2582803260, 8135830269, 25569752274, 80196041223, 251048476872, 784526490225, 2447722649502

Offset: 1

Author

Danail Bonchev (bonchevd(AT)aol.com), Sep 07 2001

Keywords

• nonn
• nice
• easy

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

Examples

			a(5) = 162 because we can write (5+1)*3^(5-2) = 6*3^3 = 6*27.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..200
• Index entries for linear recurrences with constant coefficients, signature (6,-9).

Crossrefs

Cf. A001764, A006234, A014915, A025192, A027261, A079272, A196389.

Programs

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 *)

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

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).

a(n) = A025192(n) + A027471(n). (End)

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)