A209919 Triangle read by rows: T(n,k), 0 <= k <= n-1, = number of 2-divided binary sequences of length n which are 2-divisible in exactly k ways.

0, 3, 1, 4, 2, 2, 6, 3, 4, 3, 8, 6, 6, 6, 6, 14, 9, 11, 10, 11, 9, 20, 18, 18, 18, 18, 18, 18, 36, 30, 33, 30, 34, 30, 33, 30, 60, 56, 56, 58, 56, 56, 58, 56, 56, 108, 99, 105, 99, 105, 100, 105, 99, 105, 99, 188, 186, 186, 186, 186, 186, 186, 186, 186, 186, 186, 352, 335, 344, 338, 346, 335, 348, 335, 346, 338, 344, 335, 632, 630, 630, 630, 630, 630, 630, 630, 630, 630, 630, 630, 630, 1182, 1161, 1179, 1161, 1179, 1161, 1179, 1162, 1179, 1161, 1179, 1161, 1179, 1161, 2192, 2182, 2182, 2188, 2182, 2184, 2188, 2182, 2182, 2188, 2184, 2182, 2188, 2182, 2182

Offset

1,2

Comments

Computed by David Scambler.

See A210109 for further information.

Omitting the leading column, triangle has mirror symmetry.

Speculation: T(2n+1,2)=T(2n+1,1); T(2n,2)=T(2n,1)+T(n,1); T(3n+1,3)=T(3n+1,1); T(3n+2,3)=T(3n+2,1); T(3n,3)=T(3n,1)+T(n,1) and similar "lagged modulo sums" for T(4n+i,4)=T(4n+i,2), 0<i<=3; T(4n,4)=T(4n,2)+T(n,1); T(5n+i,5)=T(5n+i,1), 0<i<=4; T(5n,5)=T(5n,1)+T(n,1). - R. J. Mathar, Mar 27 2012

Right border appears to be A059966. - Michel Marcus, Apr 26 2013

Links

Table of n, a(n) for n=1..120.

Example

Triangle begins:
n  k=0  k=1  k=2  k=3  k=4  k=5  k=6  k=7  k=8  k=9  k=10 k=11 k=12 k=13 k=14
1  1
2  3    1
3  4    2    2
4  6    3    4    3
5  8    6    6    6    6
6  14   9    11   10   11   9
7  20   18   18   18   18   18   18
8  36   30   33   30   34   30   33   30
9  60   56   56   58   56   56   58   56   56
10 108  99   105  99   105  100  105  99   105  99
11 188  186  186  186  186  186  186  186  186  186  186
12 352  335  344  338  346  335  348  335  346  338  344  335
13 632  630  630  630  630  630  630  630  630  630  630  630  630
14 1182 1161 1179 1161 1179 1161 1179 1162 1179 1161 1179 1161 1179 1161
15 2192 2182 2182 2188 2182 2184 2188 2182 2182 2188 2184 2182 2188 2182 2182...

Crossrefs

First column is A000031, second column is conjectured to be A001037. Row sums = 2^n.

Sequence in context: A237016 A135822 A242111 * A116537 A194307 A346770

Adjacent sequences: A209916 A209917 A209918 * A209920 A209921 A209922

Keyword

nonn,tabl

Author

N. J. A. Sloane, Mar 21 2012

Status

approved