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A245163 T(n,k)=Number of length n 0..k arrays with new values introduced in order from both ends.

%I A245163 #6 Jul 23 2025 11:29:28
%S A245163 1,1,1,1,1,2,1,1,2,4,1,1,2,4,8,1,1,2,4,9,16,1,1,2,4,9,23,32,1,1,2,4,9,
%T A245163 23,64,64,1,1,2,4,9,23,65,186,128,1,1,2,4,9,23,65,199,551,256,1,1,2,4,
%U A245163 9,23,65,199,653,1645,512,1,1,2,4,9,23,65,199,654,2275,4926,1024,1,1,2,4,9,23
%N A245163 T(n,k)=Number of length n 0..k arrays with new values introduced in order from both ends.
%C A245163 Table starts
%C A245163 .....1........1.........1.........1.........1.........1.........1.........1
%C A245163 .....1........1.........1.........1.........1.........1.........1.........1
%C A245163 .....2........2.........2.........2.........2.........2.........2.........2
%C A245163 .....4........4.........4.........4.........4.........4.........4.........4
%C A245163 .....8........9.........9.........9.........9.........9.........9.........9
%C A245163 ....16.......23........23........23........23........23........23........23
%C A245163 ....32.......64........65........65........65........65........65........65
%C A245163 ....64......186.......199.......199.......199.......199.......199.......199
%C A245163 ...128......551.......653.......654.......654.......654.......654.......654
%C A245163 ...256.....1645......2275......2296......2296......2296......2296......2296
%C A245163 ...512.....4926......8313......8568......8569......8569......8569......8569
%C A245163 ..1024....14768.....31439.....33794.....33825.....33825.....33825.....33825
%C A245163 ..2048....44293....121637....140039....140580....140581....140581....140581
%C A245163 ..4096...132867....477307....605869....612890....612933....612933....612933
%C A245163 ..8192...398588...1888721...2718531...2794159...2795181...2795182...2795182
%C A245163 .16384..1195750...7509799..12564289..13280627..13298407..13298464..13298464
%C A245163 .32768..3587235..29940861..59419764..65597882..65851100..65852872..65852873
%C A245163 .65536.10761689.119550419.285878342.335521900.338654554.338694406.338694479
%H A245163 R. H. Hardin, <a href="/A245163/b245163.txt">Table of n, a(n) for n = 1..9999</a>
%F A245163 Empirical for column k:
%F A245163 k=1: a(n) = 2*a(n-1) for n>2
%F A245163 k=2: a(n) = 5*a(n-1) -7*a(n-2) +3*a(n-3) for n>4
%F A245163 k=3: a(n) = 10*a(n-1) -37*a(n-2) +64*a(n-3) -52*a(n-4) +16*a(n-5) for n>6
%F A245163 k=4: [order 7] for n>8
%F A245163 k=5: [order 9] for n>10
%F A245163 k=6: [order 11] for n>12
%F A245163 k=7: [order 13] for n>14
%e A245163 Some solutions for n=10 k=4
%e A245163 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A245163 ..1....1....1....1....0....1....0....1....0....1....1....0....0....1....1....0
%e A245163 ..1....0....2....0....1....2....1....0....0....2....2....0....1....1....2....1
%e A245163 ..0....1....3....1....2....0....0....2....0....2....1....1....0....1....2....0
%e A245163 ..1....2....0....0....0....1....2....3....0....3....1....2....1....1....2....1
%e A245163 ..0....2....0....2....2....1....0....3....1....1....0....1....2....2....2....0
%e A245163 ..0....1....2....1....2....1....0....2....1....3....2....1....1....1....1....2
%e A245163 ..0....1....1....2....1....1....1....0....1....2....1....1....1....2....1....1
%e A245163 ..1....1....1....1....1....1....0....1....0....1....1....0....1....1....1....1
%e A245163 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%Y A245163 Column 1 is A000079(n-2)
%Y A245163 Column 2 is A164039(n-2)
%Y A245163 Diagonal is A007476
%K A245163 nonn,tabl
%O A245163 1,6
%A A245163 _R. H. Hardin_, Jul 12 2014