cp's OEIS Frontend

A348666 a(n) is the number of quiddities of 3-periodic dissections of (n + 2)-gons.

%I A348666 #17 Feb 02 2022 07:25:09
%S A348666 1,1,2,5,15,49,166,577,2050,7414,27201,100984,378651,1431901,5454718,
%T A348666 20912754,80630085,312430832,1216045522,4752132953,18638125275,
%U A348666 73340870891,289463959745,1145612705905,4545478673125,18077348646721,72048928923617,287733587217552,1151233484320195
%N A348666 a(n) is the number of quiddities of 3-periodic dissections of (n + 2)-gons.
%C A348666 See Conley-Ovsienko paper, p. 6.
%C A348666 a(0) = 1 by convention.
%H A348666 Vaclav Kotesovec, <a href="/A348666/b348666.txt">Table of n, a(n) for n = 0..1000</a>
%H A348666 Charles H. Conley and Valentin Ovsienko, <a href="https://arxiv.org/abs/2107.01234">Quiddities of polygon dissections and the Conway-Coxeter frieze equation</a>, arXiv:2107.01234 [math.CO], 2021.
%H A348666 Charles H Conley and Valentin Ovsienko, <a href="https://arxiv.org/abs/2202.00269">Counting quiddities of polygon dissections</a>, arXiv:2202.00269 [math.CO], 2021.
%H A348666 Vaclav Kotesovec, <a href="/A348666/a348666.txt">Recurrence (of order 12)</a>
%F A348666 a(n) = Sum_{k=0..n/3} Sum_{s=0..k} ((3*(k-s) + 2)/(n-s+1)) * binomial(n-3*k+s-2,s) * binomial(2*n-3*k-s-1,n-3*k-1).
%F A348666 a(n) ~ c * d^n / n^(3/2), where d = 4.21429839439676340483426656814177802445... is the root of the equation 4 - 12*d^2 - 8*d^3 + 12*d^4 - 20*d^5 + d^7 = 0 and c = 0.590856549086828350357357054105900401452384216047617779361986537... - _Vaclav Kotesovec_, Nov 04 2021
%t A348666 {1}~Join~Array[Sum[(3 (k - s) + 2)/(# - s + 1)*Binomial[# - 3 k + s - 2, s]*Binomial[2 # - 3 k - s - 1, # - 3 k - 1], {k, 0, #/3}, {s, 0, k}] &, 29]
%Y A348666 Cf. A218251.
%K A348666 nonn,easy
%O A348666 0,3
%A A348666 _Michael De Vlieger_, Oct 28 2021