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

1, 1, 2, 5, 15, 49, 166, 577, 2050, 7414, 27201, 100984, 378651, 1431901, 5454718, 20912754, 80630085, 312430832, 1216045522, 4752132953, 18638125275, 73340870891, 289463959745, 1145612705905, 4545478673125, 18077348646721, 72048928923617, 287733587217552, 1151233484320195

Offset

0,3

Comments

See Conley-Ovsienko paper, p. 6.

a(0) = 1 by convention.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Charles H. Conley and Valentin Ovsienko, Quiddities of polygon dissections and the Conway-Coxeter frieze equation, arXiv:2107.01234 [math.CO], 2021.

Charles H Conley and Valentin Ovsienko, Counting quiddities of polygon dissections, arXiv:2202.00269 [math.CO], 2021.

Vaclav Kotesovec, Recurrence (of order 12)

Formula

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).

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

Mathematica

{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]

Crossrefs

Cf. A218251.

Sequence in context: A149932 A149933 A149934 * A149935 A149936 A189176

Adjacent sequences: A348663 A348664 A348665 * A348667 A348668 A348669

Keyword

nonn,easy

Author

Michael De Vlieger, Oct 28 2021

Status

approved