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A100092 Number of n-celled polyominoes with minimum perimeter. 6
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
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

S. Kurz, Counting polyominoes with minimum perimeter, submitted to Ars Combinatoria

S. Kurz, Counting polyominoes with minimum perimeter, arXiv:math/0506428 [math.CO], 2005-2015.

Kival Ngaokrajang, Illustration of initial terms ["A275966" should be changed to "A100092"]

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-Fran├žois Alcover, Jul 20 2018 *)

CROSSREFS

Cf. A027709, A100093, A100094, left nonzero term in row n of A342243.

Sequence in context: A252733 A181876 A131505 * A131508 A305502 A308211

Adjacent sequences:  A100089 A100090 A100091 * A100093 A100094 A100095

KEYWORD

nonn

AUTHOR

Sascha Kurz, Nov 03 2004

EXTENSIONS

Offset changed to 0 by N. J. A. Sloane, Mar 19 2017

STATUS

approved

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Last modified August 15 21:23 EDT 2022. Contains 356148 sequences. (Running on oeis4.)