A393655 - OEIS

A393655

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.

35, 153, 603, 2241, 8019, 27945, 95499, 321489, 1069443, 3523257, 11514555, 37378017, 120637107, 387420489, 1238788971, 3945949425, 12526595811, 39646030041, 125136817947, 394006637313, 1237808462355, 3880791038313, 12144470068683, 37939701067281, 118337975785539, 368570545107705

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OFFSET

3,1

COMMENTS

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.

That is, a(n) equals the total number of descents over all such paths from a fixed vertex on level 2*(n+1).

LINKS

Tamilselvi Annamalai, Table of n, a(n) for n = 3..102

Mantharam Parvathi, Annamalai Tamilselvi, and Devanbu Hepsi, p^(k)-Fibonacci Numbers of the p-Bratteli Diagram for Every Odd Prime p and Integer k>=0, arXiv:2603.04229 [math.CO], 2026.

Wikipedia, Bratteli diagram.

Index entries for linear recurrences with constant coefficients, signature (6,-9).

FORMULA

a(n) = 3^(n-3)*(16*n-13).

G.f.: x^3*(35-57*x)/(1-3*x)^2.

E.g.f.: (26 - 18*x - 171*x^2 + exp(3*x)*(96*x - 26))/54. - Stefano Spezia, Feb 25 2026

EXAMPLE

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.

Grouping:

4 paths with 3 descents: 4*3=12

9 paths with 2 descents: 9*2=18

5 paths with 1 descent: 5*1=5

Hence, a(3)=12+18+5=35.

MAPLE

a(n):= 3^(n - 3)*(16*n - 13);

num_list := [seq(eval(a(n), n = i), i = 3 .. 100)];

MATHEMATICA

LinearRecurrence[{6, -9}, {35, 153}, 28] (* Hugo Pfoertner, Feb 24 2026 *)

CROSSREFS

Cf. A391520, A393661, A393662.