growing binary trees

[Ralf Stephan]

> I made up an interesting sequence:
> 
> 1, 2, 3, 4, 6, 8, 10, 13, 17, 21, 26, 33, 41,
> 50, 62, 77, 94, 115, 142, 174, 212, 260, 319,
> 389, 475, 582, 711, 867, 1060, 1296

According to superseeker:

                                     2      3      4
                        6 + 2 x + 2 x  - 3 x  - 4 x
                     [- ----------------------------, ogf]
                                5            3
                              -x  - 1 + x + x

Second cumulative sums of:

%I A017817
%S A017817 1,0,0,1,1,0,1,2,1,1,3,3,2,4,6,5,6,10,11,11,16,21,22,27,37,43,49,64,80,
%T A017817 92,113,144,172,205,257,316,377,462,573,693,839,1035,1266,1532,1874,
%U A017817 2301,2798,3406,4175,5099,6204,7581,9274,11303,13785,16855,20577,25088
%N A017817 a(0)=1, a(1)=a(2)=0, a(3)=1; a(n)=a(n-3)+a(n-4).
%F A017817 G.f.: 1/(1-x^3-x^4).
%F A017817 a(n)/a(n-1) tends to r = 1.2207440846..., a real root of x^4 - x - 1 = 0. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 22 2006

First sums of:

Transformation T018 gave a match with:
%I A122511
%S A122511 1,1,1,2,2,2,3,4,4,5,7,8,9,12,15,17,21,27,32,38,48,59,70,86,107,129,156,
%T A122511 193,236,285,349,429,521,634,778,950,1155,1412,1728,2105,2567,3140,3833,
%U A122511 4672,5707,6973,8505,10379,12680,15478
%N A122511 Alternative method for A079398 using vector matrirx Markov.
%K A122511 nonn,uned
%A A122511 Roger Bagula (rlbagula(AT)sbcglobal.net), Sep 16 2006

which is IDENTICAL with

%I A079398
%S A079398 0,1,1,1,1,2,2,2,3,4,4,5,7,8,9,12,15,17,21,27,32,38,48,59,70,86,107,129,
%T A079398 156,193,236,285,349,429,521,634,778,950,1155,1412,1728,2105,2567,3140,
%U A079398 3833,4672,5707,6973,8505,10379,12680,15478,18884,23059,28158,34362
%N A079398 Conjectured values of d(n), the dimension of a Z-module in MZV(conv).
%C A079398 From the conjecture of Zagier, Drinfeld, Kontsevich and Goncharov (see link).
%C A079398 P(0)=P(1)=P(2)=P(3)=1, for m>3: P(m) = P(m-3) + P(m-4) is the 3rd sequence in the series: Fibonacci sequence, Padovan sequence, ... The Padovan sequence (whose ratio of successive terms approaches the plastic constant) is similar to the Perrin sequence. - Jonathan Vos Post (jvospost2(AT)yahoo.com), Jan 23 2005
%D A079398 Michel Waldschmidt, "Multiple Zeta values and Euler-Zagier numbers", in Number theory and discrete mathematics, International conference in honour of Srinivasa Ramanujan, Center for Advanced Study in Mathematics, Panjab University, Chandigarh, (Oct 02, 2000).
%H A079398 Michel Waldschmidt, <a href="http://www.institut.math.jussieu.fr/~miw/articles/pdf/Transparents.pdf">Multiple Zeta values and Euler-Zagier numbers</a>
%H A079398 Eric W. Weisstein, <a href="http://mathworld.wolfram.com/PadovanSequence.html">Padovan Sequence</a>.
%F A079398 a(1)=0 a(2)=a(3)=a(4)=1 for n>=4 a(n)=a(n-2)+a(n-3)
%F A079398 a(n)=sum{k=0..floor((n-1)/2), binomial(floor((n-k-1)/3), k)} (offset 0); a(n)=sum{k=0..floor(n/2), binomial(floor((n-k-1)/3), k)}-0^n. (offset 0). - Paul Barry (pbarry(AT)wit.ie), Jul 06 2004
%F A079398 For n>1, a(n) = P(n-2) where P(n) is defined by: P(0)=P(1)=P(2)=P(3)=1, for m>3: P(m) = P(m-3) + P(m-4). - Jonathan Vos Post (jvospost2(AT)yahoo.com), Jan 23 2005
%Y A079398 Cf. A000931.
%K A079398 nonn
%O A079398 1,6
%A A079398 Benoit Cloitre (abmt(AT)wanadoo.fr), Feb 16 2003


ralf