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A100092 Number of n-celled polyominoes with minimum perimeter.

%I A100092 #38 Mar 08 2024 09:45:08
%S A100092 1,1,1,2,1,1,1,4,2,1,6,1,1,11,4,2,1,11,6,1,1,28,11,4,2,1,35,11,6,1,1,
%T A100092 65,28,11,4,2,1,73,35,11,6,1,1,147,65,28,11,4,2,1,182,73,35,11,6,1,1,
%U A100092 321,147,65,28,11,4,2,1,374,182,73,35,11,6,1,1,678,321,147,65,28,11,4,2,1
%N A100092 Number of n-celled polyominoes with minimum perimeter.
%C A100092 An n-celled polyomino has minimum perimeter A027709(n) = 2*ceiling(2*sqrt(n)). - _Dmitry Kamenetsky_, Feb 27 2017
%H A100092 Joerg Arndt, <a href="/A100092/b100092.txt">Table of n, a(n) for n = 0..144</a>
%H A100092 Sascha Kurz, <a href="http://www.mathe2.uni-bayreuth.de/sascha/papers/extremal_polyominoes_paper1_2.pdf">Counting polyominoes with minimum perimeter</a>, submitted to Ars Combinatoria
%H A100092 Sascha Kurz, <a href="https://arxiv.org/abs/math/0506428">Counting polyominoes with minimum perimeter</a>, arXiv:math/0506428 [math.CO], 2005-2015.
%H A100092 Kival Ngaokrajang, <a href="/A100092/a100092.pdf">Illustration of initial terms</a> ["A275966" should be changed to "A100092"]
%H A100092 Piotr Pikul, <a href="https://doi.org/10.1080/0025570X.2024.2312866">Tightest Arrangements of All Different Nets of a Cube</a>, Math. Mag. (2024).
%F A100092 It seems that for m >= 1, 0 <= k <= m-1, we have a(m^2-k) = a(k^2+k+1) = A100094(k) and a(m^2+m-k) = a((k+1)^2+1) = A100093(k+1). If this is true, then a(n) = 1 if and only if n is of the form m^2, m^2 + m - 1 or m^2 + m. - _Jianing Song_, Aug 10 2021
%e A100092 a(9) = 1 because the 3 X 3 square is the unique polyomino with minimum perimeter.
%t A100092 (* Warning: some local maxima are precomputed from A100094. *)
%t A100092 A100094 = {2, 4, 11, 28, 65, 147, 321, 678, 1382, 2738, 5289 (* extend if needed *)};
%t A100092 amax = Last[A100094]; nmax = 144;
%t A100092 S[x_] := 1 + Sum[ x^(2*n + 1)*Product[ (x^(2*k - 1) - 1), {k, n}], {n, 0, nmax}] + O[x]^nmax;
%t A100092 A[x_] = Product[1/(1 - x^k), {k, 1, nmax}] + O[x]^nmax // Normal;
%t A100092 R[x_] := 1/4 (A[x]^4 + 3A[x^2]^2) + O[x]^nmax;
%t A100092 Q[x_] := 1/8 (A[x]^4 + 3A[x^2]^2 + 2S[x]^2 A[x^2] + 2A[x^4]) + O[x]^nmax;
%t A100092 r[k_] := SeriesCoefficient[R[x], {x, 0, k}];
%t A100092 q[k_] := SeriesCoefficient[Q[x], {x, 0, k}];
%t A100092 e[n_] := Module[{s, w}, s = Floor[Sqrt[n]]; a94Q[k_] := IntegerQ[w = Sqrt[k + n] - k] && w > 0; Which[Evaluate[Sequence @@ Flatten[Table[{a94Q[k], A100094[[k]]}, {k, 3, Length[A100094]}]]], n == s^2, 1, IntegerQ[t = n - s^2] && 0 < t < s, Sum[r[s - c - c^2 - t], {c, 0, Floor[-1/2 + (1/2)* Sqrt[1 + 4 s - 4 t]]}], n == s^2 + s, 1, IntegerQ[t = n - s^2 - s] && 0 < t <= s, q[s + 1 - t] + Sum[r[s + 1 - c^2 - t], {c, 1, Floor[Sqrt[s + 1 - t]]}], True, Print["error n = ", n]]];
%t A100092 Select[Table[e[n], {n, 0, nmax}], # <= amax&] (* _Jean-François Alcover_, Jul 20 2018 *)
%Y A100092 Cf. A027709, A100093, A100094, left nonzero term in row n of A342243.
%K A100092 nonn
%O A100092 0,4
%A A100092 _Sascha Kurz_, Nov 03 2004
%E A100092 Offset changed to 0 by _N. J. A. Sloane_, Mar 19 2017