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A098487 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1<=k<=m positions can be picked in an m X m square array such that all positions are mutually isolated. Two positions (s,t),(u,v) are considered as isolated from each other if min(abs(s-u),abs(t-v))>1.

%I A098487 #23 Apr 26 2025 15:20:48
%S A098487 1,4,0,9,16,8,16,78,140,79,25,228,964,1987,1974,36,520,3920,16834,
%T A098487 42368,62266,49,1020,11860,85275,397014,1220298,2484382,64,1806,29708,
%U A098487 317471,2326320,12033330,44601420,119138166,81,2968,65240,962089,10087628,77784658,450193818,1979541332,6655170642
%N A098487 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1<=k<=m positions can be picked in an m X m square array such that all positions are mutually isolated. Two positions (s,t),(u,v) are considered as isolated from each other if min(abs(s-u),abs(t-v))>1.
%C A098487 For more information, links, programs see A098485.
%H A098487 Alois P. Heinz, <a href="/A098487/b098487.txt">Rows n = 1..21, flattened</a>
%e A098487 T(3,3) = a(6) = 8 because there are the following 8 ways to pick 3 positions isolated from each other from a 3 X 3 square array:
%e A098487 X0X...X0X...X0X...X00...X00...0X0...00X...00X
%e A098487 000...000...000...00X...000...000...X00...000
%e A098487 X00...0X0...00X...X00...X0X...X0X...00X...X0X
%e A098487 Triangle begins:
%e A098487 :  1;
%e A098487 :  4,    0;
%e A098487 :  9,   16,     8;
%e A098487 : 16,   78,   140,     79;
%e A098487 : 25,  228,   964,   1987,    1974;
%e A098487 : 36,  520,  3920,  16834,   42368,    62266;
%e A098487 : 49, 1020, 11860,  85275,  397014,  1220298,  2484382;
%e A098487 : 64, 1806, 29708, 317471, 2326320, 12033330, 44601420, 119138166;
%o A098487 (Fortran) ! See link in A098485.
%Y A098487 A098485 gives selections where all marks are connected, A090642 gives total number of possible selections.
%Y A098487 Main diagonal gives A201513.
%Y A098487 Cf. A291716, A291717, A291718, A292152, A292153, A292154, A292155, A292156.
%K A098487 nonn,tabl
%O A098487 1,2
%A A098487 _Hugo Pfoertner_, Sep 15 2004
%E A098487 T(8,8) corrected by _Alois P. Heinz_, May 11 2017