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%I #43 Mar 19 2026 09:42:40
%S 35,153,603,2241,8019,27945,95499,321489,1069443,3523257,11514555,
%T 37378017,120637107,387420489,1238788971,3945949425,12526595811,
%U 39646030041,125136817947,394006637313,1237808462355,3880791038313,12144470068683,37939701067281,118337975785539,368570545107705
%N Total number of descents over all paths from a fixed vertex on level 2*(n+1) to level 1 in the 3-Bratteli diagram, a(n) = 3^(n-3)*(16*n-13), n>=3.
%C For each n>=0, a(n) = Sum_{P} des(P), where the sum runs over all directed paths from level 2*(n+1) to level 1 in the 3-Bratteli diagram.
%C That is, a(n) equals the total number of descents over all such paths from a fixed vertex on level 2*(n+1).
%H Tamilselvi Annamalai, <a href="/A393655/b393655.txt">Table of n, a(n) for n = 3..102</a>
%H Mantharam Parvathi, Annamalai Tamilselvi, and Devanbu Hepsi, <a href="https://arxiv.org/abs/2603.04229">p^(k)-Fibonacci Numbers of the p-Bratteli Diagram for Every Odd Prime p and Integer k>=0</a>, arXiv:2603.04229 [math.CO], 2026.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bratteli_diagram">Bratteli diagram</a>.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9).
%F a(n) = 3^(n-3)*(16*n-13).
%F G.f.: x^3*(35-57*x)/(1-3*x)^2.
%F E.g.f.: (26 - 18*x - 171*x^2 + exp(3*x)*(96*x - 26))/54. - _Stefano Spezia_, Feb 25 2026
%e For n=3, there are 18 paths from level 2*(3+1) to level 1. The numbers of descents in these 18 paths are: 3, 3, 2, 3, 2, 3, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 1, 1.
%e Grouping:
%e 4 paths with 3 descents: 4*3=12
%e 9 paths with 2 descents: 9*2=18
%e 5 paths with 1 descent: 5*1=5
%e Hence, a(3)=12+18+5=35.
%p a(n):= 3^(n - 3)*(16*n - 13);
%p num_list := [seq(eval(a(n), n = i), i = 3 .. 100)];
%t LinearRecurrence[{6, -9}, {35, 153}, 28] (* _Hugo Pfoertner_, Feb 24 2026 *)
%Y Cf. A391520, A393661, A393662.
%K nonn,easy
%O 3,1
%A _Tamilselvi Annamalai_, Feb 24 2026
%E a(79) corrected and b-file index shifted by _Georg Fischer_, Mar 19 2026