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A209919
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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.
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3
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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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...
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CROSSREFS
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First column is A000031, second column is conjectured to be A001037. Row sums = 2^n.
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KEYWORD
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AUTHOR
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STATUS
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approved
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