OFFSET
1,5
COMMENTS
The n-th row contains (n-1)^2 + 1 elements.
LINKS
Alois P. Heinz, Rows n = 1..16, flattened (Rows n = 1..7 from Christopher Hunt Gribble)
FORMULA
Sum_{k=0..(n-1)^2} T(n,k) = A045846(n).
From Christopher Hunt Gribble, Jul 02 2013: (Start)
It appears that:
T(n,1) = (n-1)^2, n>1 = A000290(n-1).
T(n,2) = (n-2)(n-3)(n^2+n-4)/2, n>2 = A061995(n-1).
T(n,3) = (n-2)(n-3)(n^4-n^3-23n^2+15n+140)/6, n>2 = A061996(n-1).
T(n,4) = (n^8 - 8n^7 - 26*n^6 + 340*n^5 - 105*n^4 - 4708*n^3 + 6814*n^2 + 20852*n - 40248)/24, n>3. (End)
EXAMPLE
For n = 3, there are 4 tilings that contain 1 isolated node, so T(3,1) = 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 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
The irregular triangle begins:
\ k 0 1 2 3 4 5 6 7 8 9 ...
n
1 1
2 1 1
3 1 4 0 0 1
4 1 9 16 8 5 0 0 0 0 1
5 1 16 78 140 88 44 68 32 0 4 ...
6 1 25 228 964 2003 2178 1842 1626 725 290 ...
7 1 36 520 3920 16859 42944 67312 72980 69741 62952 ...
MAPLE
b:= proc(n, l) option remember; local i, k, s, t;
if max(l[])>n then 0 elif n=0 or l=[] then 1
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:=0;
for i from k to nops(l) while l[i]=0 do s:=s+x^((i-k)^2)
*b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)])
od; expand(s)
fi
end:
T:= n-> (l-> seq(coeff(l, x, i), i=0..degree(l)))(b(n, [0$n])):
seq(T(n), n=1..9); # Alois P. Heinz, Jun 27 2013
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Christopher Hunt Gribble, Jun 26 2013
STATUS
approved