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

1,1

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

Links

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

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

Example

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

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).

Sequence in context: A114027 A005877 A144174 * A098354 A354455 A209382

Adjacent sequences: A256813 A256814 A256815 * A256817 A256818 A256819

Keyword

nonn,tabl

Author

R. H. Hardin, Apr 10 2015

Status

approved