A255992 T(n,k)=Number of length n+k 0..1 arrays with at most one downstep in every k consecutive neighbor pairs.

4, 8, 8, 15, 16, 16, 26, 28, 32, 32, 42, 45, 53, 64, 64, 64, 68, 80, 100, 128, 128, 93, 98, 114, 144, 188, 256, 256, 130, 136, 156, 196, 256, 354, 512, 512, 176, 183, 207, 257, 337, 451, 667, 1024, 1024, 232, 240, 268, 328, 428, 568, 796, 1256, 2048, 2048, 299, 308

Offset

1,1

Links

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

Sela Fried, A closed form formula for A255992, 2026.

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) -a(n-2) +2*a(n-3) -a(n-4)

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

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

k=6: a(n) = 2*a(n-1) -a(n-2) +5*a(n-6) -4*a(n-7)

k=7: a(n) = 2*a(n-1) -a(n-2) +6*a(n-7) -5*a(n-8)

Empirical for row n:

n=1: a(n) = (1/6)*n^3 + (1/2)*n^2 + (4/3)*n + 2

n=2: a(n) = (1/6)*n^3 + n^2 + (23/6)*n + 3

n=3: a(n) = (1/6)*n^3 + (3/2)*n^2 + (31/3)*n + 4

n=4: a(n) = (1/6)*n^3 + 2*n^2 + (143/6)*n + 6 for n>2

n=5: a(n) = (1/6)*n^3 + (5/2)*n^2 + (145/3)*n + 12 for n>3

n=6: a(n) = (1/6)*n^3 + 3*n^2 + (533/6)*n + 28 for n>4

n=7: a(n) = (1/6)*n^3 + (7/2)*n^2 + (454/3)*n + 64 for n>5

T(n,k) = n+k+1 + Sum_{r=1..floor((n-2)/k)+2} Sum_{j=0..r-1} (-1)^j*C(r-1,j)*(k-1)^(r-1-j)*(k-2)^j*C(n+(2-r)*k+2*r-1-j,2*r+1). This confirms all conjectures here and in the related sequences A255993-A255998 (see Fried link). - Sela Fried, Mar 20 2026

Example

Table starts
....4....8...15...26...42...64...93..130..176..232..299..378..470..576...697
....8...16...28...45...68...98..136..183..240..308..388..481..588..710...848
...16...32...53...80..114..156..207..268..340..424..521..632..758..900..1059
...32...64..100..144..196..257..328..410..504..611..732..868.1020.1189..1376
...64..128..188..256..337..428..530..644..771..912.1068.1240.1429.1636..1862
..128..256..354..451..568..705..854.1016.1192.1383.1590.1814.2056.2317..2598
..256..512..667..796..945.1134.1352.1584.1831.2094.2374.2672.2989.3326..3684
..512.1024.1256.1413.1574.1797.2088.2419.2766.3130.3512.3913.4334.4776..5240
.1024.2048.2365.2510.2645.2848.3175.3606.4090.4592.5113.5654.6216.6800..7407
.2048.4096.4454.4448.4476.4560.4824.5294.5912.6598.7304.8031.8780.9552.10348
Some solutions for n=4 k=4
..1....1....0....0....0....0....0....1....0....0....1....0....1....0....0....0
..1....0....0....1....1....0....0....1....1....0....1....0....1....0....0....1
..1....0....1....1....1....0....1....0....0....0....1....1....0....1....0....1
..1....1....0....1....0....1....1....0....0....0....0....1....0....0....1....1
..0....1....0....0....0....1....1....1....0....1....1....1....1....0....1....1
..1....1....0....1....1....0....1....1....0....0....1....1....1....0....1....1
..1....0....0....1....1....1....1....1....1....1....1....0....1....0....1....0
..1....1....1....1....0....1....0....1....0....1....0....1....0....0....1....1

Crossrefs

Column 1 is A000079(n+1)

Column 2 is A000079(n+2)

Column 3 is A118870(n+3)

Row 1 is A000125(n+1)

Sequence in context: A333288 A159786 A083744 * A273572 A273779 A114027

Adjacent sequences: A255989 A255990 A255991 * A255993 A255994 A255995

Keyword

nonn,tabl

Author

R. H. Hardin, Mar 13 2015

Status

approved