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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A085311 A052273 A074912 * A158502 A215244 A195427

Adjacent sequences:  A274204 A274205 A274206 * A274208 A274209 A274210

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Jun 13 2016

STATUS

approved

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Last modified April 25 13:31 EDT 2019. Contains 322461 sequences. (Running on oeis4.)