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A242322 T(n,k)=Number of length n+k+2 0..k arrays with every value 0..k appearing at least once in every consecutive k+3 elements, and new values 0..k introduced in order.

%I A242322 #6 Jul 23 2025 11:16:24
%S A242322 7,25,13,65,61,24,140,185,145,44,266,440,503,337,81,462,896,1300,1316,
%T A242322 781,149,750,1638,2801,3648,3398,1829,274,1155,2766,5334,8231,10012,
%U A242322 8801,4269,504,1705,4395,9290,16194,23486,27368,23069,9957,927,2431,6655
%N A242322 T(n,k)=Number of length n+k+2 0..k arrays with every value 0..k appearing at least once in every consecutive k+3 elements, and new values 0..k introduced in order.
%C A242322 Table starts
%C A242322 ....7....25.....65.....140.....266.....462......750.....1155.....1705.....2431
%C A242322 ...13....61....185.....440.....896....1638.....2766.....4395.....6655.....9691
%C A242322 ...24...145....503....1300....2801....5334.....9290....15123....23350....34551
%C A242322 ...44...337...1316....3648....8231...16194....28897....47931....75118...112511
%C A242322 ...81...781...3398...10012...23486...47466....86381...145443...230647...348771
%C A242322 ..149..1829...8801...27368...66366..137166...253674...432331...692113..1054531
%C A242322 ..274..4269..23069...75236..187671..395166...740496..1274419..2055676..3150991
%C A242322 ..504..9957..60197..208976..533801.1141290..2161503..3749211..6083896..9369751
%C A242322 ..927.23233.156887..577964.1530356.3312546..6326951.11042115.18002245.27827211
%C A242322 .1705.54225.408962.1596216.4371836.9669270.18590776.32600811.53341987.82686971
%H A242322 R. H. Hardin, <a href="/A242322/b242322.txt">Table of n, a(n) for n = 1..369</a>
%F A242322 Empirical for column k:
%F A242322 k=1: a(n) = a(n-1) +a(n-2) +a(n-3)
%F A242322 k=2: a(n) = a(n-1) +2*a(n-2) +2*a(n-3) +2*a(n-4) -a(n-5) -a(n-6)
%F A242322 k=3: [order 10]
%F A242322 k=4: [order 15]
%F A242322 k=5: [order 21]
%F A242322 k=6: [order 28]
%F A242322 Empirical for row n:
%F A242322 n=1: a(n) = (1/8)*n^4 + (11/12)*n^3 + (19/8)*n^2 + (31/12)*n + 1
%F A242322 n=2: a(n) = (5/8)*n^4 + (35/12)*n^3 + (39/8)*n^2 + (43/12)*n + 1
%F A242322 n=3: a(n) = (21/8)*n^4 + (89/12)*n^3 + (67/8)*n^2 + (55/12)*n + 1
%F A242322 n=4: a(n) = (77/8)*n^4 + (179/12)*n^3 + (103/8)*n^2 + (67/12)*n + 1
%F A242322 n=5: a(n) = (261/8)*n^4 + (245/12)*n^3 + (163/8)*n^2 + (79/12)*n + 1
%F A242322 n=6: a(n) = (845/8)*n^4 - (73/12)*n^3 + (343/8)*n^2 + (91/12)*n + 1 for n>1
%F A242322 n=7: a(n) = (2661/8)*n^4 - (2263/12)*n^3 + (1059/8)*n^2 + (103/12)*n + 1 for n>2
%F A242322 n=8: a(n) = (8237/8)*n^4 - (11701/12)*n^3 + (3879/8)*n^2 + (115/12)*n + 1 for n>3
%F A242322 n=9: a(n) = (25221/8)*n^4 - (46531/12)*n^3 + (14275/8)*n^2 + (127/12)*n + 1 for n>4
%F A242322 n=10: a(n) = (76685/8)*n^4 - (165601/12)*n^3 + (50455/8)*n^2 + (139/12)*n + 1 for n>5
%F A242322 n=11: a(n) = (232101/8)*n^4 - (555055/12)*n^3 + (171139/8)*n^2 + (151/12)*n + 1 for n>6
%F A242322 n=12: a(n) = (700397/8)*n^4 - (1794061/12)*n^3 + (561703/8)*n^2 + (163/12)*n + 1 for n>7
%F A242322 n=13: a(n) = (2109381/8)*n^4 - (5664667/12)*n^3 + (1798755/8)*n^2 + (175/12)*n + 1 for n>8
%F A242322 n=14: a(n) = (6344525/8)*n^4 - (17608249/12)*n^3 + (5657175/8)*n^2 + (187/12)*n + 1 for n>9
%F A242322 n=15: a(n) = (19066341/8)*n^4 - (54151687/12)*n^3 + (17559907/8)*n^2 + (199/12)*n + 1 for n>10
%e A242322 Some solutions for n=5 k=4
%e A242322 ..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
%e A242322 ..1....1....0....1....1....1....1....1....1....1....1....0....1....0....1....1
%e A242322 ..0....2....1....1....2....2....1....2....2....2....1....0....2....1....0....1
%e A242322 ..2....1....2....2....3....2....0....1....3....3....2....1....3....2....2....2
%e A242322 ..3....3....3....3....0....3....2....3....1....4....3....2....4....0....0....0
%e A242322 ..4....0....0....0....2....2....3....4....0....2....2....3....3....3....3....3
%e A242322 ..1....4....4....4....4....4....4....0....4....0....4....4....2....4....4....4
%e A242322 ..2....1....2....4....1....0....1....2....3....3....0....1....0....1....4....4
%e A242322 ..0....1....0....1....1....1....1....3....3....1....1....2....1....1....1....1
%e A242322 ..3....2....1....2....3....4....0....1....2....1....4....0....3....3....2....2
%e A242322 ..1....0....4....0....2....4....0....1....2....2....4....4....2....2....0....4
%Y A242322 Column 1 is A000073(n+5)
%Y A242322 Row 1 is A001296(n+1)
%K A242322 nonn,tabl
%O A242322 1,1
%A A242322 _R. H. Hardin_, May 10 2014