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%I #34 Jan 28 2023 12:01:26
%S 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,
%T 13,14,15,14,15,14,15,16,15,16,15,16,17,16,17,16,17,18,17,18,17,18,19,
%U 18,19,18,19,18,19,20,19,20,19,20,21,20,21,20,21,20,21,22,21,22,21,22
%N Minimal perimeter of polyiamond with n triangles.
%C A polyiamond is a shape made up of n congruent equilateral triangles.
%D Frank Harary and Heiko Harborth, Extremal animals, J. Combinatorics Information Syst. Sci., 1(1):1-8, 1976.
%H Stefano Spezia, <a href="/A067628/b067628.txt">Table of n, a(n) for n = 0..10000</a>
%H Greg Malen, Érika Roldán, and Rosemberg Toalá-Enríquez, <a href="https://doi.org/10.1007/s00026-022-00631-1">Extremal {p, q}-Animals</a>, Ann. Comb. (2023). See Corollary 1.9 at p. 8.
%H Greg Malen and Érika Roldán, <a href="https://arxiv.org/abs/1906.08447">Polyiamonds Attaining Extremal Topological Properties</a>, arXiv:1906.08447 [math.CO], 2019.
%H J. Yackel, R. R. Meyer, and I. Christou, <a href="https://doi.org/10.1007/BF02614375">Minimum-perimeter domain assignment</a>, Mathematical Programming, vol. 78 (1997), pp. 283-303.
%H W. C. Yang and R. R. Meyer, <a href="http://digital.library.wisc.edu/1793/64366">Maximal and minimal polyiamonds</a>, 2002.
%F Let c(n) = ceiling(sqrt(6n)). Then a(n) is whichever of c(n) or c(n) + 1 has the same parity as n.
%F a(n) = 2*ceiling((n + sqrt(6*n))/2) - n (Harary and Harborth, 1976). - _Stefano Spezia_, Oct 02 2019
%p 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;
%o (PARI) a(n)=2*ceil((n+sqrt(6*n))/2)-n; \\ _Stefano Spezia_, Oct 02 2019
%o (Python)
%o from math import isqrt
%o def A067628(n): return (c:=isqrt(6*n-1)+1)+((c^n)&1) if n else 0 # _Chai Wah Wu_, Jul 28 2022
%Y Cf. A000105, A000577, A027709 (squares), A057729, A135711.
%K nonn
%O 0,2
%A Winston C. Yang (winston(AT)cs.wisc.edu), Feb 02 2002
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