

A103469


Number of polyominoes consisting of 3 regular unit ngons.


13



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
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OFFSET

3,2


COMMENTS

Conjecture: if n > 3, then a(n + 3) is the number of connected components of the subgraph that is vertexinduced 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



FORMULA

a(n) = floor((n2)/2)  floor((n1)/6) + 1.
G.f.: x^3*(x^6x^5+x^4x^3x1) / ((x1)^2*(x+1)*(x^2x+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^6x^5+x^4x^3x1)/((x1)^2*(x+1)*(x^2x+1)*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Jan 19 2015


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



