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

Author

R. H. Hardin, May 08 2014

Keywords

• nonn
• tabl

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

Examples

			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
		

Links

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

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

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)