A067628 Minimal perimeter of polyiamond with n triangles.

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

Author

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

Keywords

• nonn

Comments

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

References

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

Links

• 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.

Crossrefs

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

Programs

Maple

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;

PARI

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

Python

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

Formula

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