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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
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
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).
Sequence in context: A052273 A369291 A074912 * A158502 A331813 A215244
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jun 13 2016
STATUS
approved