A103469 Number of polyominoes consisting of 3 regular unit n-gons.

1, 2, 2, 3, 2, 3, 3, 4, 4, 5, 4, 5, 5, 6, 6, 7, 6, 7, 7, 8, 8, 9, 8, 9, 9, 10, 10, 11, 10, 11, 11, 12, 12, 13, 12, 13, 13, 14, 14, 15, 14, 15, 15, 16, 16, 17, 16, 17, 17, 18, 18, 19, 18, 19, 19, 20, 20, 21, 20, 21, 21, 22, 22, 23, 22, 23, 23, 24, 24, 25, 24, 25, 25, 26, 26, 27, 26, 27

Offset

3,2

Comments

Conjecture: if n > 3, then a(n + 3) is the number of connected components of the subgraph that is vertex-induced on Collatz's graph by the vertex subset {1, ..., n} (see Problem 3.11 of my article, available from the links). - Lorenzo Sauras Altuzarra, Apr 07 2020 [Corrected by Pontus von Brömssen, Jan 22 2021]

Links

Colin Barker, Table of n, a(n) for n = 3..1000

M. Koch and S. Kurz, Enumeration of generalized polyominoes, arXiv:math/0605144 [math.CO], 2006.

S. Kurz, k-polyominoes.

Lorenzo Sauras Altuzarra, Some arithmetical problems that are obtained by analyzing proofs and infinite graphs, arXiv:2002.03075 [math.NT], 2020.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Formula

a(n) = floor((n-2)/2) - floor((n-1)/6) + 1.

G.f.: -x^3*(x^6-x^5+x^4-x^3-x-1) / ((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1)). - Colin Barker, Jan 19 2015

Example

a(3)=1 because there is only one polyiamond consisting of 3 triangles and a(4)=2 because there are 2 polyominoes consisting of 3 squares.

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 1, -1}, {1, 2, 2, 3, 2, 3, 3}, 80] (* Harvey P. Dale, Sep 18 2016 *)

Prog

(PARI) Vec(-x^3*(x^6-x^5+x^4-x^3-x-1)/((x-1)^2*(x+1)*(x^2-x+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Jan 19 2015

Crossrefs

Cf. A103470, A103471, A103472, A103473.

Sequence in context: A282630 A108309 A341307 * A337932 A029326 A239500

Adjacent sequences: A103466 A103467 A103468 * A103470 A103471 A103472

Keyword

nonn,easy

Author

Sascha Kurz, Feb 07 2005

Status

approved