A264071 T(n,k) = number of (n+1) X (k+1) arrays of permutations of 0..(n+1)*(k+1)-1 with each element having index change +-(.,.) 0,0 1,1 or 1,2.

2, 5, 4, 13, 21, 8, 34, 121, 89, 16, 89, 605, 1210, 377, 32, 233, 3025, 12100, 12100, 1597, 64, 610, 15125, 131890, 239580, 121000, 6765, 128, 1597, 75625, 1445345, 5645376, 4745620, 1210000, 28657, 256, 4181, 378125, 15892745, 130697424, 242621698, 94000060, 12100000, 121393, 512

Offset: 1

Author

R. H. Hardin, Nov 02 2015

Keywords

• nonn
• tabl

Examples

			Table starts:
     2       5          13             34                89                  233
     4      21         121            605              3025                15125
     8      89        1210          12100            131890              1445345
    16     377       12100         239580           5645376            130697424
    32    1597      121000        4745620         242621698          11909009849
    64    6765     1210000       94000060       10427064769        1084282319384
   128   28657    12100000     1861931060      448121165789       98725402363225
   256  121393   121000000    36880691100    19258783041289     8989061417123964
   512  514229  1210000000   730524027860   827679549612058   818464496640651553
  1024 2178309 12100000000 14470047586940 35570961850254336 74522143720797473932
  ...
Some solutions for n=3 k=4:
..0..8..9..3..4....6..1..2..3..4....7..8..2..3..4....0..8..2..3..4
..5.13..7..1..2...11.13..0..8..9...12..6..0..1..9....5..6..7..1..9
.16.11.18..6.14...10..5.19..7.14...17..5.18.13.14...17.18.12.13.14
.15.10.17.12.19...15.16.17.18.12...15.16.10.11.19...15.16.10.11.19
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..143

Crossrefs

Main diagonal is A264066.

Columns 1,2 and 4..7 are A000079, A015448(n+1), A264067, A264068, A264069, A264070.

Rows 1 and 3..7 are A001519(n+1), A264072, A264073, A264074, A264075, A264076.

Formula

Empirical for column k:

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

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

k=3: a(n) = 10*a(n-1) for n>2

k=4: a(n) = 19*a(n-1) +16*a(n-2) for n>3

k=5: a(n) = 43*a(n-1) -43*a(n-3) +a(n-4) for n>5

k=6: a(n) = 87*a(n-1) +374*a(n-2) -470*a(n-3) +207*a(n-4) +3*a(n-5) for n>7

k=7: a(n) = 191*a(n-1) +1102*a(n-2) -7594*a(n-3) -38349*a(n-4) +38507*a(n-5) for n>8

Empirical for row n:

n=1: a(n) = 3*a(n-1) -a(n-2)

n=2: a(n) = 5*a(n-1) for n>3

n=3: a(n) = 12*a(n-1) -12*a(n-2) +12*a(n-3) -12*a(n-4) +a(n-5) for n>7

n=4: a(n) = 24*a(n-1) -19*a(n-2) -11*a(n-3) +36*a(n-4) +3*a(n-5) for n>9

n=5: [order 14] for n>19

n=6: [order 10] for n>19

Extensions

Name corrected by Andrew Howroyd, Mar 25 2025