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%I #18 Apr 27 2022 08:55:43
%S 0,1,5,34,386,6940,221672,12582472,1293374998,242394178200,
%T 83374069529638,52845726291860344,61928161880183204434,
%U 134499571879749571406816,542432658409586214809714176,4068438590479352629770422328000,56820656114941381799512710314429768
%N Number of squares in all tilings of an n X n square using integer-sided square tiles.
%p b:= proc(n, l) option remember; local i, k, s, t;
%p if max(l[])>n then [0, 0] elif n=0 or l=[] then [1, 0]
%p elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
%p else for k do if l[k]=0 then break fi od; s:=[0$2];
%p for i from k to nops(l) while l[i]=0 do s:=s+(h->h+[0, h[1]])
%p (b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
%p od; s
%p fi
%p end:
%p a:= n-> b(n, [0$n])[2]:
%p seq(a(n), n=0..10);
%t b[n_, l_] := b[n, l] = Module[{i, k, s, t},
%t Which[Max[l] > n, {0, 0}, n == 0 || l == {}, {1, 0},
%t Min[l] > 0, t = Min[l]; b[n - t, l - t], True,
%t k = Position[l, 0, 1][[1, 1]]; s = {0, 0};
%t For[i = k, i <= Length[l] && l[[i]] == 0, i++,
%t s = s + Function[h, h + {0, h[[1]]}][b[n, Join[l[[1;; k-1]],
%t Table[1+i-k, {j, k, i}], l[[i+1;;]]]]]]; s]];
%t a[n_] := b[n, Array[0&, n]][[2]];
%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 15}] (* _Jean-François Alcover_, Apr 27 2022, after _Alois P. Heinz_ in A226545 *)
%Y Main diagonal of A226545.
%Y Row sums of A226936.
%Y Cf. A045846.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Jun 10 2013
%E a(16) from _Alois P. Heinz_, Nov 16 2016
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