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A064017
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Number of ternary trees (A001764) with n nodes and maximal diameter.
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8
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| A problem important for polymer science because it counts the trees having unbranched branches; they are called "combs".
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 31 2010: (Start)
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 (End)
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LINKS
| Harry J. Smith, Table of n, a(n) for n = 1..200
Index to sequences with linear recurrences with constant coefficients, signature (6,-9).
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FORMULA
| a(n) = 3*a(n-1)+3^(n-2); closed formula: (n+1)*3^(n-2).
a(n)=(n+2)3^(n-1)+0^n/3 (offset 0); a(n)=A025192(n)+A027471(n). - Paul Barry (pbarry(AT)wit.ie), Sep 05 2003
A006234(n+4) - a(n+2) = 3^n - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Mar 01 2005
a(n+1)= Sum_{0<=k<=n} A196389(n,k)*3^k. - From DELEHAM Philippe, Oct 31 2011
G.f.: (1-3*x+3*x^2)*x/(1-3*x)^2. - From DELEHAM Philippe, Oct 31 2011
a(1)=1, a(2)=3, a(3)=12, a(n)=6*a(n-1)-9*a(n-2) [From Harvey P. Dale, Feb 07 2012]
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EXAMPLE
| a(5)=162 because we write (5+1)*3^(5-2)=6*3^3=6*27
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MAPLE
| a:=n->ceil(sum(3^(n-2), j=0..n)): seq(a(n), n=1..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 05 2008
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MATHEMATICA
| Join[{1}, Table[(n+1)3^(n-2), {n, 2, 30}]] (* or *) Join[{1}, LinearRecurrence[ {6, -9}, {3, 12}, 30]] (* From Harvey P. Dale, Feb 07 2012 *)
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PROG
| Floretion Algebra Multiplication Program, FAMP Code: lesforseq[ - 'i + 'j - 'kk' - 'ki' - 'kj' ], vesforseq(n) = 3^n, tesforseq = A006234
(PARI) { for (n=1, 200, if (n>1, a=(n + 1)*p; p*=3, a=p=1); write("b064017.txt", n, " ", a) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Sep 06 2009]
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CROSSREFS
| Cf. A001764, A014915, A027261, A079272, A006234.
Sequence in context: A190051 A109437 A005656 * A005320 A062561 A128593
Adjacent sequences: A064014 A064015 A064016 * A064018 A064019 A064020
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KEYWORD
| nonn,nice,easy,changed
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AUTHOR
| Danail Bonchev (bonchevd(AT)aol.com), Sep 07 2001
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