A057729 -id:A057729 - cp's OEIS frontend

A284869 Number of n-step 2-dimensional closed self-avoiding paths on triangular lattice, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

0, 0, 1, 1, 1, 4, 5, 16, 37, 120, 344, 1175, 3807, 13224, 45645, 161705, 575325, 2074088, 7521818, 27502445, 101134999, 374128188

Offset: 1

Luca Petrone, Apr 04 2017

Differs from A057729 beginning at n = 11, since that sequence includes triangular polyominoes with holes.

a(n) is the number of simply connected polyiamonds with perimeter n. - Walter Trump, Nov 29 2023

Rade Doroslovački, Ivan Stojmenović and Ratko Tošić, Generating and counting triangular systems, BIT Numerical Mathematics, 27 (1987), 18-24. See Table 1.
Hugo Pfoertner, Illustration of ratio A036418(n)/a(n) using Plot2.
Hugo Pfoertner, Illustration of polygons of perimeter <= 11.
Walter Trump, Self-avoiding closed walks on a triangular lattice

Approaches (1/12)*A036418 for increasing n.

Cf. A057729, A258206, A266549, A316196.

a(15) from Hugo Pfoertner, Jun 27 2018

a(16)-a(22) from Walter Trump, Nov 29 2023

A067628 Minimal perimeter of polyiamond with n triangles.

Original entry on oeis.org

0, 3, 4, 5, 6, 7, 6, 7, 8, 9, 8, 9, 10, 9, 10, 11, 10, 11, 12, 11, 12, 13, 12, 13, 12, 13, 14, 13, 14, 15, 14, 15, 14, 15, 16, 15, 16, 15, 16, 17, 16, 17, 16, 17, 18, 17, 18, 17, 18, 19, 18, 19, 18, 19, 18, 19, 20, 19, 20, 19, 20, 21, 20, 21, 20, 21, 20, 21, 22, 21, 22, 21, 22

Offset: 0

Winston C. Yang (winston(AT)cs.wisc.edu), Feb 02 2002

nonn

A polyiamond is a shape made up of n congruent equilateral triangles.

Frank Harary and Heiko Harborth, Extremal animals, J. Combinatorics Information Syst. Sci., 1(1):1-8, 1976.

Stefano Spezia, Table of n, a(n) for n = 0..10000
Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, Extremal {p, q}-Animals, Ann. Comb. (2023). See Corollary 1.9 at p. 8.
Greg Malen and Érika Roldán, Polyiamonds Attaining Extremal Topological Properties, arXiv:1906.08447 [math.CO], 2019.
J. Yackel, R. R. Meyer, and I. Christou, Minimum-perimeter domain assignment, Mathematical Programming, vol. 78 (1997), pp. 283-303.
W. C. Yang and R. R. Meyer, Maximal and minimal polyiamonds, 2002.

Cf. A000105, A000577, A027709 (squares), A057729, A135711.

interface(quiet=true); for n from 0 to 100 do if (1 = 1) then temp1 := ceil(sqrt(6*n)); end if; if ((temp1 mod 2) = (n mod 2)) then temp2 := 0; else temp2 := 1; end if; printf("%d,", temp1 + temp2); od;

a(n)=2*ceil((n+sqrt(6*n))/2)-n; \\ Stefano Spezia, Oct 02 2019

from math import isqrt
def A067628(n): return (c:=isqrt(6*n-1)+1)+((c^n)&1) if n else 0 # Chai Wah Wu, Jul 28 2022

Let c(n) = ceiling(sqrt(6n)). Then a(n) is whichever of c(n) or c(n) + 1 has the same parity as n.

a(n) = 2*ceiling((n + sqrt(6*n))/2) - n (Harary and Harborth, 1976). - Stefano Spezia, Oct 02 2019

A069813 Maximum number of triangles in polyiamond with perimeter n.

Original entry on oeis.org

1, 2, 3, 6, 7, 10, 13, 16, 19, 24, 27, 32, 37, 42, 47, 54, 59, 66, 73, 80, 87, 96, 103, 112, 121, 130, 139, 150, 159, 170, 181, 192, 203, 216, 227, 240, 253, 266, 279, 294, 307, 322, 337, 352, 367, 384, 399, 416, 433, 450, 467, 486, 503, 522, 541, 560, 579

Offset: 3

Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002

			a(10) = 16 because the maximum number of triangles in a polyiamond of perimeter 10 is 16.

Colin Barker, Table of n, a(n) for n = 3..1000
W. C. Yang, R. R. Meyer, Maximal and minimal polyiamonds, 2002.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A000105, A000577, A027709, A030511 (bisection), A057729, A067628.

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( x^3*(x^2-x-1)*(x^2+1)/((x-1)^3*(x+1)*(x^2+x+1)))); // Marius A. Burtea, Jan 03 2020

A069813 := proc(n)
    round(n^2/6) ;
    if modp(n,6) <> 0 then
        %-1 ;
    else
        % ;
    end if;
end proc: # R. J. Mathar, Jul 14 2015

LinearRecurrence[{1, 1, 0, -1, -1, 1}, {1, 2, 3, 6, 7, 10}, 60] (* Jean-François Alcover, Jan 03 2020 *)

a(n) = round(n^2/6) - (n % 6 != 0) \\ Michel Marcus, Jul 17 2013

Vec(x^3*(x^2-x-1)*(x^2+1)/((x-1)^3*(x+1)*(x^2+x+1)) + O(x^60)) \\ Colin Barker, Jan 19 2015

a(n) = round(n^2/6) - (0 if n = 0 mod 6, 1 else) = A056829(n)-A097325(n).
From Colin Barker, Jan 18 2015: (Start)
a(n) = round((-25 + 9*(-1)^n + 8*exp(-2/3*i*n*Pi) + 8*exp((2*i*n*Pi)/3) + 6*n^2)/36), where i=sqrt(-1).
G.f.: x^3*(1+x-x^2)*(1+x^2) / ((1-x)^3*(1+x)*(1+x+x^2)). (End)
a(n) = A001399(n-3) + A001399(n-4) + A001399(n-6) - A001399(n-7). - R. J. Mathar, Jul 14 2015

A131481 a(n) is the number of n-cell polyiamonds (triangular polyominoes) with perimeter n+2.

Original entry on oeis.org

1, 1, 1, 3, 4, 11, 23, 62, 149, 409, 1066, 2931, 7981, 22166, 61508, 172267, 483088, 1361475, 3845139, 10894630

Offset: 1

Tanya Khovanova, Jul 27 2007

n+2 is the maximal perimeter for an n-celled polyiamond. a(n) is the number of n-celled polyiamonds that have a tree as their connectedness graph (vertices of this graph correspond to cells and two vertices are connected if the corresponding cells have a common edge)

Andrew Clarke, Isoperimetrical Polyiamonds

a(n) <= A000577(n), a(n) <= A057729(n+2).

Offset corrected and terms a(17)-a(20) added by John Mason, May 15 2021

A260666 Number of patterns with perimeter n in the planar net 3.3.4.3.4, mirrors and holes are excluded.

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 6, 13

Offset: 1

Kival Ngaokrajang, Nov 14 2015

Inspired by A057729 which is in planar net 3.3.3.3.3.3.

Kival Ngaokrajang, Illustration of initial terms

Cf. A057729.

A387209 Number of convex polygons with perimeter n on the regular triangular lattice, not counting rotations and reflections as distinct.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 2, 4, 4, 6, 5, 10, 7, 12, 11, 16, 13, 22, 17, 26, 23, 32, 27, 41, 33, 47, 42, 56, 48, 68, 57, 77, 69, 89, 78, 105, 90, 117, 106, 133, 118, 153, 134, 169, 154, 189, 170, 214, 190, 234, 215, 259, 235, 289, 260, 314, 290, 344, 315, 380

Offset: 0

Walter Trump, Aug 22 2025

a(n) also is the number of convex polyiamonds (triangular polyominoes) with perimeter n.

Walter Trump, Table of n, a(n) for n = 0..10000
Walter Trump, Convex Polyiamonds
Walter Trump, Examples for a(3)-a(12)

Cf. A096004 (number of convex polyiamonds with n cells), A284869 (including nonconvex but simply connected polyiamonds with perimeter n), A057729 (including polyiamonds with holes), A036418 (including rotations and reflections but no holes).

A130616 Number of triangular polyominoes (or polyiamonds) with perimeter at most n.

Original entry on oeis.org

0, 0, 1, 2, 3, 7, 12, 28, 65, 185

Offset: 1

Tanya Khovanova, Aug 10 2007

Partial sums of A057729 - Number of triangular polyominoes (or polyiamonds) with perimeter n. a(n+2) >= A130867(n) - Triangular polyominoes (or polyiamonds) with n cells at most.

Cf. A131481 (number of n-cell polyiamonds with perimeter n+2).

Offset corrected by John Mason, Jan 16 2023

A131486 a(n) is the number of triangular polyominoes (polyiamonds) with n edges.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 4, 1, 11, 1, 23, 4, 62, 11, 150, 38, 411, 118, 1081, 389

Offset: 1

Tanya Khovanova, Jul 28 2007

An n-celled polyiamond with perimeter p has (3n+p)/2 edges. The maximum number of edges in an n-celled polyiamond is 2n+1.

Isoperimetrical Polyiamonds at Poly Pages

Cf. A000577: Triangular polyominoes (or polyiamonds) with n cells. A057729: Number of triangular polyominoes (or polyiamonds) [A000577] with perimeter n. A131481: a(n) is the number of n-cell polyiamonds (triangular polyominoes) with perimeter n+2.

cp's OEIS Frontend

A284869 Number of n-step 2-dimensional closed self-avoiding paths on triangular lattice, reduced for symmetry, i.e., where rotations and reflections are not counted as distinct.

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A067628 Minimal perimeter of polyiamond with n triangles.

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A069813 Maximum number of triangles in polyiamond with perimeter n.

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A131481 a(n) is the number of n-cell polyiamonds (triangular polyominoes) with perimeter n+2.

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A260666 Number of patterns with perimeter n in the planar net 3.3.4.3.4, mirrors and holes are excluded.

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A387209 Number of convex polygons with perimeter n on the regular triangular lattice, not counting rotations and reflections as distinct.

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A130616 Number of triangular polyominoes (or polyiamonds) with perimeter at most n.

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A131486 a(n) is the number of triangular polyominoes (polyiamonds) with n edges.

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