A273346 -id:A273346 - cp's OEIS frontend

A273347 The sum of the areas of the bargraphs of semiperimeter n (n>=2).

1, 4, 16, 60, 218, 778, 2744, 9600, 33391, 115638, 399137, 1374050, 4720272, 16187632, 55434424, 189607406, 647872199, 2211794804, 7545239986, 25722676402, 87641357150, 298456587038, 1015914399813, 3456670564220, 11757143968393, 39976448714086, 135887220346719

Offset: 2

Emeric Deutsch, Jun 03 2016

nonn

			a(4) = 16  because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and, clearly, the sum of their areas is 3 + 3 + 3 + 4 + 3 = 16.

A. Blecher, C. Brennan, and A. Knopfmacher, Combinatorial parameters in bargraphs (preprint).

Alois P. Heinz, Table of n, a(n) for n = 2..1000
M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
M. Bousquet-Mélou and R. Brak, Exactly solved models of polyominoes and polygons, Chapter 3 of Polygons, Polyominoes and Polycubes, Lecture Notes in Physics, Vol. 775, 43-78, Springer, Berlin, Heidelberg 2009.
Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Cf. A273346, A273348.

Q := sqrt(1-4*z+2*z^2+z^4): g := z^2*(2-z-z^3-z*Q)^2/(1-4*z+z^4+Q+z^2*(2+Q))^2; gser:= series(g,z = 0,40): seq(coeff(gser, z, m), m = 2 .. 35);
# second Maple program:
a:= proc(n) option remember;
      `if`(n<7, [0$2, 1, 4, 16, 60, 218, 778][n+1],
      ((69-115*n+28*n^2)*a(n-1) -(264-265*n+52*n^2)*a(n-2)
       +(3*(29-29*n+4*n^2))*a(n-3) -(3*(10-21*n+4*n^2))*a(n-4)
       +(4*n-9)*(5*n-29)*a(n-5) +(4*n-13)*(n-6)*a(n-6)
       +(n-7)*(4*n-9)*a(n-7))/(n*(4*n-13)))
    end:
seq(a(n), n=2..30);  # Alois P. Heinz, Jun 04 2016

b[n_, y_, t_] := b[n, y, t] = Expand[If[n==0, 1-t, If[t<0, 0, b[n-1, y+1, 1]] + If[t>0 || y<2, 0, b[n, y-1, -1]] + If[y<1, 0, b[n-1, y, 0]*z^y]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 1, Exponent[p, z]}]][b[n, 0, 0]]; a[n_] := (row = T[n]; row.Range[Length[row]]); Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Nov 29 2016 after Alois P. Heinz's Maple code for A273346 *)

G.f.: g(z) = z^2 (2 - z - z^3 - z q )^2 /(1 - 4z + z^4 + q + z^2 q)^2, where q = sqrt(1 - 4z + 2z^2 + z^4) (see Section 4.3 of the Blecher et al. reference).
a(n) = Sum(k*A273346(n,k), k>=1).
a(n) = ((69-115*n+28*n^2)*a(n-1) -(264-265*n+52*n^2)*a(n-2) +(3*(29-29*n+4*n^2))*a(n-3) -(3*(10-21*n+4*n^2))*a(n-4) +(4*n-9) * (5*n-29)*a(n-5) +(4*n-13)*(n-6)*a(n-6) +(n-7)*(4*n-9)*a(n-7))/ (n*(4*n-13)) for n>=7. - Alois P. Heinz, Jun 04 2016

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.

Original entry on oeis.org

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

Alois P. Heinz, Jun 13 2016

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.

			              _
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

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

Row sums give A075126.
Column sums give A000079(k-1).
Cf. A001787, A273346, A274208, A274217.

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);

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 *)

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).

A273348 The sum of the semiperimeters of the bargraphs of area n (n>=1).

Original entry on oeis.org

2, 6, 16, 39, 92, 211, 476, 1059, 2332, 5091, 11036, 23779, 50972, 108771, 231196, 489699, 1034012, 2177251, 4572956, 9582819, 20039452, 41826531, 87148316, 181287139, 376555292, 781072611, 1618069276, 3347986659, 6919669532, 14286731491, 29468247836, 60726065379, 125031270172, 257220819171, 528758195996

Offset: 1

Emeric Deutsch, Jun 03 2016

nonn

			a(4) = 39 because the 8 bargraphs of area 4 correspond to the compositions [2,2],[4],[3,1],[1,3],[2,1,1],[1,2,1],[1,1,2],[1,1,1,1] and the sum of their semiperimeters is 4 + 7*5 = 39.

A. Blecher, C. Brennan, and A. Knopfmacher, Combinatorial parameters in bargraphs, Quaestiones Mathematicae, Volume 39, 2016 - Issue 5.
M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
M. Bousquet-Mélou and R. Brak, Exactly solved models of polyominoes and polygons, Chapter 3 of Polygons, Polyominoes and Polycubes, Lecture Notes in Physics, Vol. 775, 43-78, Springer, Berlin, Heidelberg 2009.
Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-4,4).

Cf. A273346, A273347.

a := proc(n) (5/12)*n*2^n+(29/36)*2^n-(1/18)*(-1)^n-1/2 end proc:
seq(a(n), n = 1 .. 35);

LinearRecurrence[{4, -3, -4, 4}, {2, 6, 16, 39}, 35] (* Jean-François Alcover, Nov 27 2017 *)

first(n) = Vec(x*(2-2*x-2*x^2+x^3)/((1-x^2)*(1-2*x)^2) + O(x^(n+1))) \\ Iain Fox, Nov 27 2017

G.f.: g = t(2-2t-2t^2+t^3)/((1-t^2)(1-2t)^2).
a(n) = (15*n2^n+29*2^n-2(-1)^n-18)/36.
a(n) = Sum_{k>=2} k * A273346(k,n).

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A273347 The sum of the areas of the bargraphs of semiperimeter n (n>=2).

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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.

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A273348 The sum of the semiperimeters of the bargraphs of area n (n>=1).

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