A274207 Number T(n,k) of bargraphs of site-perimeter n having area k; triangle T(n,k), n>=4, floor((n-1)/2)<=k<=floor(((n-1)^2+3)/12), read by rows.

1, 2, 2, 2, 2, 4, 4, 2, 4, 7, 1, 6, 6, 10, 4, 2, 9, 13, 14, 12, 2, 8, 13, 22, 18, 24, 10, 2, 2, 15, 27, 40, 29, 38, 28, 12, 2, 10, 24, 45, 65, 59, 58, 56, 40, 16, 4, 2, 23, 52, 84, 104, 112, 100, 95, 88, 56, 28, 7, 1, 12, 40, 92, 148, 181, 205, 191, 172, 163, 132, 96, 48, 16, 4

Offset

4,2

Comments

A bargraph is a polyomino whose bottom is a segment of the nonnegative x-axis and whose upper part is a lattice path starting at (0,0) and ending with its first return to the x-axis using steps U=(0,1), D=(0,-1) and H=(1,0), where UD and DU are not allowed.

The site-perimeter of a polyomino is the number of exterior cells having a common edge with at least one polyomino cell.

Links

Alois P. Heinz, Rows n = 4..100, flattened

M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Wikipedia, Polyomino

Formula

Sum_{k=floor((n-1)/2)..floor(((n-1)^2+3)/12)} k * T(n,k) = A274208(n).

Sum_{n>=4} k * T(n,k) = A001787(k).

Sum_{n>=4} n * T(n,k) = A274217(k).

Example

              _
T(4,1) = 1:  |_|
              _
             | |     ___
T(6,2) = 2:  |_|    |___|
              _        _
             | |_    _| |
T(7,3) = 2:  |___|  |___|
              _
             | |
             | |     _____
T(8,3) = 2:  |_|    |_____|
              ___      _
             |   |   _| |_
T(8,4) = 2:  |___|  |_____|
              _        _
             | |      | |   _            _
             | |_    _| |  | |___    ___| |
T(9,4) = 4:  |___|  |___|  |_____|  |_____|
              _        _
             | |_    _| |   ___        ___
             |   |  |   |  |   |_    _|   |
T(9,5) = 4:  |___|  |___|  |_____|  |_____|
                _
              _| |_
             |     |
T(10,7) = 1: |_____|
.
Triangle T(n,k) begins:
n\k: 1 2 3 4 5 6  7  8  9 10  11  12  13 14 15 16 17 . .
---+----------------------------------------------------
04 : 1
05 :
06 :   2
07 :     2
08 :     2 2
09 :       4 4
10 :       2 4 7  1
11 :         6 6 10  4
12 :         2 9 13 14 12  2
13 :           8 13 22 18 24  10   2
14 :           2 15 27 40 29  38  28  12  2
15 :             10 24 45 65  59  58  56 40 16  4
16 :              2 23 52 84 104 112 100 95 88 56 28 7 1

Maple

b:= proc(n, y, t, w) option remember; `if`(n<0, 0, `if`(n=0, (1-t),
     `if`(t<0, 0, b(n-`if`(w>0 or t=0, 1, 2), y+1, 1, max(0, w-1)))+
     `if`(t>0 or y<2, 0, b(n, y-1, -1, `if`(t=0, 1, w+1))) +expand(
     `if`(y<1, 0, z^y*b(n-`if`(t<0, 1, 2), y, 0, `if`(t<0, w, 0))))))
    end:
T:= n-> (p-> seq(coeff(p, z, i),
         i= iquo(n-1, 2)..iquo((n-1)^2+3, 12)))(b(n, 0, 1, 0)):
seq(T(n), n=4..20);

Mathematica

b[n_, y_, t_, w_] := b[n, y, t, w] = If[n<0, 0, If[n==0, (1-t), If[t<0, 0, b[n - If[w>0 || t==0, 1, 2], y+1, 1, Max[0, w-1]]] + If[t>0 || y<2, 0, b[n, y-1, -1, If[t==0, 1, w+1]]] + Expand[If[y<1, 0, z^y*b[n - If[t<0, 1, 2], y, 0, If[t<0, w, 0]]]]]];
T[n_] := Function[p, Table[Coefficient[p, z, i], {i, Quotient[n-1, 2], Quotient[(n-1)^2 + 3, 12]}]][b[n, 0, 1, 0]];
Table[T[n], {n, 4, 20}] // Flatten (* Jean-François Alcover, Apr 28 2018, after Alois P. Heinz *)

Crossrefs

Row sums give A075126.

Column sums give A000079(k-1).

Cf. A001787, A273346, A274208, A274217.

Sequence in context: A052273 A369291 A074912 * A158502 A331813 A215244

Adjacent sequences: A274204 A274205 A274206 * A274208 A274209 A274210

Keyword

nonn,tabf

Author

Alois P. Heinz, Jun 13 2016

Status

approved