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A200838
T(n,k)=Number of 0..k arrays x(0..n+1) of n+2 elements without any two consecutive increases or two consecutive decreases
12
8, 25, 16, 56, 69, 32, 105, 194, 191, 64, 176, 435, 676, 529, 128, 273, 846, 1817, 2356, 1465, 256, 400, 1491, 4108, 7587, 8210, 4057, 512, 561, 2444, 8239, 19930, 31677, 28610, 11235, 1024, 760, 3789, 15128, 45465, 96690, 132263, 99700, 31113, 2048
OFFSET
1,1
COMMENTS
Table starts
....8.....25......56......105.......176........273........400.........561
...16.....69.....194......435.......846.......1491.......2444........3789
...32....191.....676.....1817......4108.......8239......15128.......25953
...64....529....2356.....7587.....19930......45465......93472......177381
..128...1465....8210....31677.....96690.....250913.....577660.....1212729
..256...4057...28610...132263....469116....1384813....3570086.....8291391
..512..11235...99700...552247...2276028....7642875...22063924....56687801
.1024..31113..347434..2305835..11042700...42181611..136360286...387572529
.2048..86161.1210736..9627715..53576350..232803603..842739040..2649819955
.4096.238605.4219166.40199277.259938722.1284861277.5208328180.18116728573
LINKS
FORMULA
Empirical for columns:
k=1: a(n) = 2*a(n-1)
k=2: a(n) = 3*a(n-1) -a(n-2) +a(n-3)
k=3: a(n) = 4*a(n-1) -2*a(n-2) +a(n-3) -a(n-4)
k=4: a(n) = 5*a(n-1) -4*a(n-2) +3*a(n-3) -3*a(n-4) +a(n-5) -a(n-6)
k=5: a(n) = 6*a(n-1) -6*a(n-2) +3*a(n-3) -5*a(n-4) +3*a(n-5) -2*a(n-6) +a(n-7)
k=6: a(n) = 7*a(n-1) -9*a(n-2) +6*a(n-3) -9*a(n-4) +7*a(n-5) -7*a(n-6) +5*a(n-7) -2*a(n-8) +a(n-9)
k=7: a(n) = 8*a(n-1) -12*a(n-2) +6*a(n-3) -10*a(n-4) +12*a(n-5) -11*a(n-6) +11*a(n-7) -6*a(n-8) +3*a(n-9) -a(n-10)
Empirical for rows:
n=1: a(k) = (2/3)*k^3 + 3*k^2 + (10/3)*k + 1
n=2: a(k) = (5/12)*k^4 + (19/6)*k^3 + (79/12)*k^2 + (29/6)*k + 1
n=3: a(k) = (4/15)*k^5 + (17/6)*k^4 + (28/3)*k^3 + (73/6)*k^2 + (32/5)*k + 1
n=4: a(k) = (61/360)*k^6 + (93/40)*k^5 + (779/72)*k^4 + (521/24)*k^3 + (1801/90)*k^2 + (239/30)*k + 1
n=5: a(k) = (34/315)*k^7 + (163/90)*k^6 + (1981/180)*k^5 + (557/18)*k^4 + (7807/180)*k^3 + (1361/45)*k^2 + (333/35)*k + 1
n=6: a(k) = (277/4032)*k^8 + (1375/1008)*k^7 + (4933/480)*k^6 + (2723/72)*k^5 + (14161/192)*k^4 + (11197/144)*k^3 + (216211/5040)*k^2 + (929/84)*k + 1
n=7: a(k) = (124/2835)*k^9 + (1123/1120)*k^8 + (244/27)*k^7 + (1991/48)*k^6 + (57133/540)*k^5 + (74183/480)*k^4 + (291427/2268)*k^3 + (9739/168)*k^2 + (568/45)*k + 1
EXAMPLE
Some solutions for n=4 k=3
..1....2....3....0....1....1....2....1....3....3....3....1....2....0....1....1
..0....0....0....2....1....0....3....3....1....3....0....3....2....3....1....0
..0....0....2....2....0....3....0....0....1....2....1....3....2....0....1....1
..3....0....1....3....3....3....3....2....1....2....0....1....2....0....0....1
..3....3....3....0....3....0....1....2....1....1....3....3....3....2....2....3
..1....3....2....0....1....3....3....2....2....1....0....1....2....1....1....0
CROSSREFS
Column 1 is A000079(n+2)
Column 2 is A098182(n+3)
Row 1 is A131423(n+1)
Sequence in context: A023056 A103954 A217012 * A302160 A122984 A254341
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin Nov 23 2011
STATUS
approved