A228461 Two-dimensional array read by antidiagonals: T(n,k) = number of arrays of maxima of three adjacent elements of some length n+2 0..k array.

2, 3, 4, 4, 9, 7, 5, 16, 22, 11, 6, 25, 50, 46, 17, 7, 36, 95, 130, 91, 27, 8, 49, 161, 295, 310, 183, 44, 9, 64, 252, 581, 821, 736, 383, 72, 10, 81, 372, 1036, 1847, 2227, 1821, 819, 117, 11, 100, 525, 1716, 3703, 5615, 6254, 4673, 1749, 189, 12, 121, 715, 2685, 6812

Offset: 1

Author

R. H. Hardin Aug 22 2013

Keywords

• nonn
• tabl

Comments

There are two arrays (or lists, or vectors) involved, a length n+2 array with free elements from 0..k (thus (k+1)^(n+2) of them) and an array that is being enumerated of length n, each element of the latter being the maximum of three adjacent elements of the first array.

Many different first arrays can give the same second array.

Examples

			Table starts
...2....3.....4.....5......6......7.......8.......9......10.......11.......12
...4....9....16....25.....36.....49......64......81.....100......121......144
...7...22....50....95....161....252.....372.....525.....715......946.....1222
..11...46...130...295....581...1036....1716....2685....4015.....5786.....8086
..17...91...310...821...1847...3703....6812...11721...19117....29843....44914
..27..183...736..2227...5615..12453...25096...46941...82699...138699...223224
..44..383..1821..6254..17487..42386...92430..185727..349558...623513..1063283
..72..819..4673.18394..57303.151882..357510..768231.1535578..2893605..5191407
.117.1749.12107.55285.194064.567835.1453506.3357985.7152815.14263777.26930773
Some solutions for n=4 k=4
..3....4....4....3....3....4....3....4....3....0....3....3....4....2....0....2
..0....4....1....3....2....0....2....4....1....0....3....3....1....2....0....0
..4....1....1....0....1....0....4....0....0....0....2....1....4....2....2....3
..4....0....3....3....2....0....4....2....3....1....3....1....4....0....4....3
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..1700

Crossrefs

Column 1 is A005252(n+3)

Column 2 is A217878

Column 3 is A217949.

A228464 is another column.

Row 1 is A000027(n+1)

Row 2 is A000290(n+1)

Row 3 is A002412(n+1)

Row 4 is A006324(n+1)

See A217883, A217954 for similar arrays.

Formula

Empirical for column k:

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

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

k=3: [order 10]

k=4: [order 13]

k=5: [order 16]

k=6: [order 19]

k=7: [order 22]

Empirical for row n:

n=1: a(n) = n + 1

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

n=3: a(n) = (2/3)*n^3 + (5/2)*n^2 + (17/6)*n + 1

n=4: a(n) = (1/3)*n^4 + 2*n^3 + (25/6)*n^2 + (7/2)*n + 1

n=5: a(n) = (2/15)*n^5 + (7/6)*n^4 + (25/6)*n^3 + (19/3)*n^2 + (21/5)*n + 1

n=6: [polynomial of degree 6]

n=7: [polynomial of degree 7]

Extensions

Edited by N. J. A. Sloane, Sep 02 2013