A228461 - OEIS

%I #18 Dec 07 2016 23:29:29

%S 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,

%T 310,183,44,9,64,252,581,821,736,383,72,10,81,372,1036,1847,2227,1821,

%U 819,117,11,100,525,1716,3703,5615,6254,4673,1749,189,12,121,715,2685,6812

%N 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.

%C 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.

%C Many different first arrays can give the same second array.

%H R. H. Hardin, <a href="/A228461/b228461.txt">Table of n, a(n) for n = 1..1700</a>

%F Empirical for column k:

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

%F 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)

%F k=3: [order 10]

%F k=4: [order 13]

%F k=5: [order 16]

%F k=6: [order 19]

%F k=7: [order 22]

%F Empirical for row n:

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

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

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

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

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

%F n=6: [polynomial of degree 6]

%F n=7: [polynomial of degree 7]

%e Table starts

%e ...2....3.....4.....5......6......7.......8.......9......10.......11.......12

%e ...4....9....16....25.....36.....49......64......81.....100......121......144

%e ...7...22....50....95....161....252.....372.....525.....715......946.....1222

%e ..11...46...130...295....581...1036....1716....2685....4015.....5786.....8086

%e ..17...91...310...821...1847...3703....6812...11721...19117....29843....44914

%e ..27..183...736..2227...5615..12453...25096...46941...82699...138699...223224

%e ..44..383..1821..6254..17487..42386...92430..185727..349558...623513..1063283

%e ..72..819..4673.18394..57303.151882..357510..768231.1535578..2893605..5191407

%e .117.1749.12107.55285.194064.567835.1453506.3357985.7152815.14263777.26930773

%e Some solutions for n=4 k=4

%e ..3....4....4....3....3....4....3....4....3....0....3....3....4....2....0....2

%e ..0....4....1....3....2....0....2....4....1....0....3....3....1....2....0....0

%e ..4....1....1....0....1....0....4....0....0....0....2....1....4....2....2....3

%e ..4....0....3....3....2....0....4....2....3....1....3....1....4....0....4....3

%Y Column 1 is A005252(n+3)

%Y Column 2 is A217878

%Y Column 3 is A217949.

%Y A228464 is another column.

%Y Row 1 is A000027(n+1)

%Y Row 2 is A000290(n+1)

%Y Row 3 is A002412(n+1)

%Y Row 4 is A006324(n+1)

%Y See A217883, A217954 for similar arrays.

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_ Aug 22 2013

%E Edited by _N. J. A. Sloane_, Sep 02 2013