A242239 T(n,k)=Number of length n+k+1 0..k arrays with every value 0..k appearing at least once in every consecutive k+2 elements, and new values 0..k introduced in order

3, 6, 5, 10, 12, 8, 15, 22, 22, 13, 21, 35, 43, 40, 21, 28, 51, 71, 82, 74, 34, 36, 70, 106, 139, 157, 136, 55, 45, 92, 148, 211, 271, 304, 250, 89, 55, 117, 197, 298, 416, 531, 586, 460, 144, 66, 145, 253, 400, 592, 821, 1047, 1129, 846, 233, 78, 176, 316, 517, 799

Offset

1,1

Comments

Table starts

...3....6...10...15....21....28....36....45....55....66....78....91...105

...5...12...22...35....51....70....92...117...145...176...210...247...287

...8...22...43...71...106...148...197...253...316...386...463...547...638

..13...40...82..139...211...298...400...517...649...796...958..1135..1327

..21...74..157..271...416...592...799..1037..1306..1606..1937..2299..2692

..34..136..304..531...821..1174..1590..2069..2611..3216..3884..4615..5409

..55..250..586.1047..1626..2332..3165..4125..5212..6426..7767..9235.10830

..89..460.1129.2059..3231..4642..6308..8229.10405.12836.15522.18463.21659

.144..846.2176.4047..6411..9256.12587.16429.20782.25646.31021.36907.43304

.233.1556.4195.7955.12716.18442.25138.32821.41527.51256.62008.73783.86581

Links

R. H. Hardin, Table of n, a(n) for n = 1..731

Formula

Empirical for column k:

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

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

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

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

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

k=6: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7)

k=7: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7) +a(n-8)

k=8: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7) +a(n-8) +a(n-9)

k=9: a(n) = a(n-1) +a(n-2) +a(n-3) +a(n-4) +a(n-5) +a(n-6) +a(n-7) +a(n-8) +a(n-9) +a(n-10)

Empirical for row n:

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

n=2: a(n) = (3/2)*n^2 + (5/2)*n + 1

n=3: a(n) = (7/2)*n^2 + (7/2)*n + 1

n=4: a(n) = (15/2)*n^2 + (9/2)*n + 1

n=5: a(n) = (31/2)*n^2 + (11/2)*n + 1 for n>1

n=6: a(n) = (63/2)*n^2 + (13/2)*n + 1 for n>2

n=7: a(n) = (127/2)*n^2 + (15/2)*n + 1 for n>3

n=8: a(n) = (255/2)*n^2 + (17/2)*n + 1 for n>4

n=9: a(n) = (511/2)*n^2 + (19/2)*n + 1 for n>5

n=10: a(n) = (1023/2)*n^2 + (21/2)*n + 1 for n>6

n=11: a(n) = (2047/2)*n^2 + (23/2)*n + 1 for n>7

n=12: a(n) = (4095/2)*n^2 + (25/2)*n + 1 for n>8

n=13: a(n) = (8191/2)*n^2 + (27/2)*n + 1 for n>9

n=14: a(n) = (16383/2)*n^2 + (29/2)*n + 1 for n>10

n=15: a(n) = (32767/2)*n^2 + (31/2)*n + 1 for n>11

Empirical large-k generalization, for k>n-4: T(n,k) = ((2^n-1)/2)*k^2 + ((2*n+1)/2)*k + 1

Empirical recurrence generalization, for column k: a(n) = sum {i in 1..k+1} a(n-i)

Example

Some solutions for n=5 k=4
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..1....1....1....1....1....1....1....1....1....1....1....1....1....0....1....1
..2....2....0....2....2....1....0....0....2....0....2....2....0....1....2....0
..1....3....2....3....3....2....2....2....3....2....3....3....2....2....3....2
..3....4....3....4....4....3....3....3....0....3....0....0....3....3....4....3
..4....0....4....1....0....4....4....4....4....4....4....4....4....4....1....4
..0....2....2....0....1....0....2....0....2....1....2....1....0....2....0....1
..2....1....1....0....2....1....1....1....1....0....1....2....1....0....2....0
..2....3....0....2....3....0....0....1....0....0....3....4....2....1....3....4
..1....0....0....3....0....2....2....2....3....2....2....3....3....1....0....2

Crossrefs

Column 1 is A000045(n+3)

Column 2 is A196700(n+3)

Row 1 is A000217(n+1)

Row 2 is A000326(n+1)

Row 3 is A069099(n+1)

Row 4 is A220083

Sequence in context: A310130 A298818 A196068 * A123089 A246978 A127780

Adjacent sequences: A242236 A242237 A242238 * A242240 A242241 A242242

Keyword

nonn,tabl

Author

R. H. Hardin, May 08 2014

Status

approved