OFFSET
0,4
COMMENTS
An n-celled polyomino has minimum perimeter A027709(n) = 2*ceiling(2*sqrt(n)). - Dmitry Kamenetsky, Feb 27 2017
LINKS
Joerg Arndt, Table of n, a(n) for n = 0..144
Sascha Kurz, Counting polyominoes with minimum perimeter, submitted to Ars Combinatoria
Sascha Kurz, Counting polyominoes with minimum perimeter, arXiv:math/0506428 [math.CO], 2005-2015.
Piotr Pikul, Tightest Arrangements of All Different Nets of a Cube, Math. Mag. (2024).
FORMULA
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
EXAMPLE
a(9) = 1 because the 3 X 3 square is the unique polyomino with minimum perimeter.
MATHEMATICA
(* Warning: some local maxima are precomputed from A100094. *)
A100094 = {2, 4, 11, 28, 65, 147, 321, 678, 1382, 2738, 5289 (* extend if needed *)};
amax = Last[A100094]; nmax = 144;
S[x_] := 1 + Sum[ x^(2*n + 1)*Product[ (x^(2*k - 1) - 1), {k, n}], {n, 0, nmax}] + O[x]^nmax;
A[x_] = Product[1/(1 - x^k), {k, 1, nmax}] + O[x]^nmax // Normal;
R[x_] := 1/4 (A[x]^4 + 3A[x^2]^2) + O[x]^nmax;
Q[x_] := 1/8 (A[x]^4 + 3A[x^2]^2 + 2S[x]^2 A[x^2] + 2A[x^4]) + O[x]^nmax;
r[k_] := SeriesCoefficient[R[x], {x, 0, k}];
q[k_] := SeriesCoefficient[Q[x], {x, 0, k}];
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]]];
Select[Table[e[n], {n, 0, nmax}], # <= amax&] (* Jean-François Alcover, Jul 20 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Sascha Kurz, Nov 03 2004
EXTENSIONS
Offset changed to 0 by N. J. A. Sloane, Mar 19 2017
STATUS
approved