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A226936 Number T(n,k) of squares of size k^2 in all tilings of an n X n square using integer-sided square tiles; triangle T(n,k), n >= 1, 1 <= k <= n, read by rows. 4
1, 4, 1, 29, 4, 1, 312, 69, 4, 1, 5598, 1184, 153, 4, 1, 176664, 40078, 4552, 373, 4, 1, 9966344, 2311632, 285414, 18160, 917, 4, 1, 1018924032, 241967774, 30278272, 2128226, 74368, 2321, 4, 1, 190191337356, 45914039784, 5860964300, 411308056, 16210982, 311784, 5933, 4, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Alois P. Heinz, Rows n = 1..15, flattened

FORMULA

Sum_{k=1..n} T(n,k) = A226554(n).

Sum_{k=1..n} k^2 * T(n,k) = n^2 * A045846(n).

EXAMPLE

For n=3 there are [29, 4, 1] squares of sizes [1^2, 2^2, 3^3] in all tilings of a 3 X 3 square:

._._._.  ._._._.  ._._._.  ._._._.  ._._._.  ._._._.

|     |  |   |_|  |_|_|_|  |_|   |  |_|_|_|  |_|_|_|

|     |  |___|_|  |   |_|  |_|___|  |_|   |  |_|_|_|

|_____|  |_|_|_|  |___|_|  |_|_|_|  |_|___|  |_|_|_|.

Triangle T(n,k) begins:

n \ k        1          2         3        4      5     6   7   8

1 :          1;

2 :          4,         1;

3 :         29,         4,        1;

4 :        312,        69,        4,       1;

5 :       5598,      1184,      153,       4,     1;

6 :     176664,     40078,     4552,     373,     4,    1;

7 :    9966344,   2311632,   285414,   18160,   917,    4,  1;

8 : 1018924032, 241967774, 30278272, 2128226, 74368, 2321,  4,  1;

MAPLE

b:= proc(n, l) option remember; local i, k, s, t;

      if max(l[])>n then [0$2] elif n=0 then [1, 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:=[0$2];

         for i from k to nops(l) while l[i]=0 do s:= s+(h->h+

           [0, h[1]*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:

T:= n-> seq(coeff(b(n, [0$n])[2], x, k), k=1..n):

seq(T(n), n=1..10);

MATHEMATICA

$RecursionLimit = 1000; b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which[Max[l] > n, {0, 0}, n == 0, {1, 0}, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1, 1][[1, 1]]; s = {0, 0}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s + Function[h, h + {0, h[[1]]*x^(1+i-k)}][b[n, Join[l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; -1]] ] ] ] ]; s] ]; T[n_] := Table[Coefficient[b[n, Array[0&, n]][[2]], x, k], {k, 1, n}]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-Fran├žois Alcover, Dec 23 2013, translated from Maple *)

CROSSREFS

Row sums give: A226554.

Main diagonal and lower diagonals give: A000012, A010709, A226892.

Cf. A045846.

Sequence in context: A134149 A035469 A290598 * A073323 A077097 A190647

Adjacent sequences:  A226933 A226934 A226935 * A226937 A226938 A226939

KEYWORD

nonn,tabl,changed

AUTHOR

Alois P. Heinz, Jun 22 2013

STATUS

approved

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Last modified September 16 11:47 EDT 2021. Contains 347472 sequences. (Running on oeis4.)