This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The 6th rows of A225777 and A225542, that is, when n = 3 and k = 3, both contain 5 entries, so a(6) = 5.
The irregular triangle begins: n,k\u 0 1 2 3 4 5 6 7 8 9 10 11 12 ... 1,1 1 2,1 1 2,2 1 1 3,1 1 3,2 1 2 3,3 1 4 0 0 1 4,1 1 4,2 1 3 1 4,3 1 6 4 0 2 4,4 1 9 16 8 5 0 0 0 0 1 5,1 1 5,2 1 4 3 5,3 1 8 12 0 3 4 5,4 1 12 37 34 15 12 4 0 0 2 5,5 1 16 78 140 88 44 68 32 0 4 0 0 0 ... ... For n = 4, k = 3, there are 4 tilings that contain 2 isolated nodes, so T(4,3,2) = 4. A 2 X 2 square contains 1 isolated node. Consider that each tiling is composed of ones and zeros where a one represents a node with one or more links to its neighbors and a zero represents a node with no links to its neighbors. Then the 4 tilings are: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
The square array starts: 1 2 3 4 5 6 7 8 9 10 11 12 ... 2 5 9 15 21 30 38 50 60 75 87 105 ... 3 9 16 31 47 73 101 142 185 244 305 386 ... 4 15 31 59 102 170 250 375 523 726 962 ... 5 21 47 102 156 307 460 711 1040 1517 ... 6 30 73 170 307 529 907 1474 2204 ... 7 38 101 250 460 907 1351 2484 ... 8 50 142 375 711 1474 2484 ... 9 60 185 523 1040 2204 ... ... A(3,2) = 9 because there are 9 parts overall in the 2 partitions of a 3 X 2 rectangle into squares with integer sides. One partition comprises 6 1 X 1 squares and the other 2 1 X 1 squares and 1 2 X 2 square giving 9 parts in total.
b:= proc(n, l) option remember; local i, k, s, t;
if max(l[])>n then {} elif n=0 or l=[] then {0}
elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
else for k do if l[k]=0 then break fi od; s:={};
for i from k to nops(l) while l[i]=0 do s:=s union
map(v->v+x^(1+i-k), b(n, [l[j]$j=1..k-1,
1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
od; s
fi
end:
A:= (n, k)-> add(coeff(add(j, j=b(max(n, k),
[0$min(n, k)])), x, i), i=1..n):
seq(seq(A(n, 1+d-n), n=1..d), d=1..15); # Alois P. Heinz, Aug 04 2013
$RecursionLimit = 1000; b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which[ Max[l]>n, {}, n == 0 || l == {}, {0}, Min[l]>0, t = Min[l]; b[n-t, l-t], True, For[k = 1, True, k++, If[l[[k]] == 0, Break[]]]; s = {}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s ~Union~ Map[Function[{v}, v + x^(1+i-k) ], b[n, Join[l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; -1]]]]]]; s]];A[n_, k_] := Sum[Coefficient[Sum[j, {j, b[Max[n, k], Array[0&, Min[n, k]]]}], x, i], {i, 1, n}]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 15}] // Flatten (* Jean-François Alcover, Mar 06 2015, after Alois P. Heinz *)
The irregular triangle T(n,k,u) begins: n,k\u 0 1 2 3 4 5 6 7 8 9 10 11 12 ... 2,1 1 3,1 1 3,2 1 1 4,1 1 4,2 1 2 1 4,3 1 2 2 0 1 5,1 1 5,2 1 2 2 5,3 1 2 4 0 2 1 5,4 1 4 13 10 6 3 1 0 0 1 6,1 1 6,2 1 3 4 1 6,3 1 3 8 3 2 3 0 0 1 6,4 1 6 23 33 24 15 6 0 2 2 1 6,5 1 6 40 101 79 74 53 13 9 11 4 0 0 ... ... T(5,3,2) = 4 because there are 4 different sets of tilings of the 5 X 3 rectangle by integer-sided squares in which each tiling contains 2 isolated nodes. Any sequence of group D2 operations will transform each tiling in a set into another in the same set. Group D2 operations are: . the identity operation . rotation by 180 degrees . reflection about a horizontal axis through the center . reflection about a vertical axis through the center A 2 X 2 square contains 1 isolated node. Consider that each tiling is composed of ones and zeros where a one represents a node with one or more links to its neighbors and a zero represents a node with no links to its neighbors. An example of a tiling in each set is: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
T(4,4,4,8) = 2 because the 4 X 4 X 4 rectangular cuboid (in this case a cube) has 2 partitions in which there are 8 nodes unconnected to any of their neighbors. The partitions are (8 2 X 2 X 2 cubes) and (37 1 X 1 X 1 cubes and 1 3 X 3 X 3 cube). The partitions and isolated nodes can be illustrated by expanding into 2 dimensions:
._______. ._______. ._______. ._______. ._______.
| | | | . | . | | | | | . | . | | | |
|___|___| |___|___| |___|___| |___|___| |___|___|
| | | | . | . | | | | | . | . | | | |
|___|___| |___|___| |___|___| |___|___| |___|___|
._______. ._______. ._______. ._______. ._______.
| |_| | . . |_| | . . |_| | |_| |_|_|_|_|
| |_| | . . |_| | . . |_| | |_| |_|_|_|_|
|_____|_| |_____|_| |_____|_| |_____|_| |_|_|_|_|
|_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_|
.
The irregular triangle begins:
u 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
n k r
1,1,1 1
2,1,1 1
2,2,1 1
2,2,2 1 1
3,1,1 1
3,2,1 1
3,2,2 1 1
3,3,1 1
3,3,2 1 1
3,3,3 1 1 0 0 0 0 0 0 1
4,1,1 1
4,2,1 1
4,2,2 1 1 1
4,3,1 1
4,3,2 1 1 1
4,3,3 1 1 1 0 0 0 0 0 1
4,4,1 1
4,4,2 1 1 1 1 1
4,4,3 1 1 1 1 1 0 0 0 1
4,4,4 1 1 1 1 1 1 1 1 2 0 0 0 0 0 0 0 0 ...
5,1,1 1
5,2,1 1
5,2,2 1 1 1
5,3,1 1
5,3,2 1 1 1
5,3,3 1 1 1 0 0 0 0 0 1 1
5,4,1 1
5,4,2 1 1 1 1 1
5,4,3 1 1 1 1 1 0 0 0 1 1 1
5,4,4 1 1 1 1 1 1 1 1 2 1 1 1 1 0 0 0 0 ...
5,5,1 1
5,5,2 1 1 1 1 1
5,5,3 1 1 1 1 1 0 0 0 1 1 1 1
5,5,4 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 0 0 ...
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