A256816 T(n,k) = Number of length n+k 0..1 arrays with at most two downsteps in every k consecutive neighbor pairs.

4, 8, 8, 16, 16, 16, 32, 32, 32, 32, 63, 64, 64, 64, 64, 120, 124, 128, 128, 128, 128, 219, 229, 245, 256, 256, 256, 256, 382, 402, 442, 484, 512, 512, 512, 512, 638, 673, 753, 856, 956, 1024, 1024, 1024, 1024, 1024, 1080, 1220, 1424, 1656, 1888, 2048, 2048, 2048

Offset: 1

Author

R. H. Hardin, Apr 10 2015

Keywords

• nonn
• tabl

Comments

Table starts

....4....8...16....32....63...120...219...382....638...1024...1586...2380

....8...16...32....64...124...229...402...673...1080...1670...2500...3638

...16...32...64...128...245...442...753..1220...1894...2836...4118...5824

...32...64..128...256...484...856..1424..2249...3402...4965...7032...9710

...64..128..256...512...956..1656..2693..4158...6153...8792..12202..16524

..128..256..512..1024..1888..3204..5088..7677..11120..15579..21230..28264

..256..512.1024..2048..3728..6192..9613.14168..20075..27566..36888..48304

..512.1024.2048..4096..7362.11955.18104.26117..36218..48738..64024..82440

.1024.2048.4096..8192.14539.23088.34013.47858..65130..86008.110976.140536

.2048.4096.8192.16384.28712.44617.63928.87338.116104.150906.191620.238932

Examples

			Some solutions for n=4, k=4
..1....1....0....0....0....0....1....0....0....0....0....0....1....0....0....1
..0....0....1....1....0....1....0....1....1....0....0....0....1....0....0....1
..1....1....0....1....0....0....1....0....1....1....1....1....0....1....0....1
..0....1....1....1....0....1....1....1....1....1....0....0....1....1....0....0
..0....1....0....0....1....1....1....1....0....0....1....1....1....1....0....0
..0....1....1....0....1....1....0....0....0....1....0....0....0....1....0....1
..0....0....1....0....1....1....1....0....0....1....1....1....0....0....0....0
..0....1....0....1....1....1....0....1....0....1....0....1....0....1....0....1
		

Links

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

Crossrefs

Column 1 is A000079(n+1).

Column 2 is A000079(n+2).

Column 3 is A000079(n+3).

Column 4 is A000079(n+4).

Row 1 is A006261(n+1).

Formula

Empirical for column k:

k=1: a(n) = 2*a(n-1)

k=2: a(n) = 2*a(n-1)

k=3: a(n) = 2*a(n-1)

k=4: a(n) = 2*a(n-1)

k=5: a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -a(n-4) +2*a(n-5) -a(n-6)

k=6: a(n) = 2*a(n-1) -a(n-2) +2*a(n-3) -a(n-4) +3*a(n-6) -2*a(n-7) -6*a(n-9) +4*a(n-10)

k=7: [order 15]

Empirical for row n:

n=1: a(n) = (1/120)*n^5 + (1/8)*n^3 + (1/2)*n^2 + (41/30)*n + 2

n=2: a(n) = (1/120)*n^5 + (1/24)*n^4 + (3/8)*n^3 - (1/24)*n^2 + (277/60)*n + 3

n=3: a(n) = (1/120)*n^5 + (1/12)*n^4 + (31/24)*n^3 - (31/12)*n^2 + (66/5)*n + 4

n=4: [polynomial of degree 5] for n>2

n=5: [polynomial of degree 5] for n>3

n=6: [polynomial of degree 5] for n>4

n=7: [polynomial of degree 5] for n>5