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.

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

Author

R. H. Hardin Nov 23 2011

Keywords

• nonn
• tabl

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

Examples

			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
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..9999

Crossrefs

Column 1 is A000079(n+2)

Column 2 is A098182(n+3)

Row 1 is A131423(n+1)

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