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A105272
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Array T(n,k) (k >= 1, n >= k) read by antidiagonals (see definition in Comments lines).
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6
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1, 1, 2, 1, 2, 2, 1, 4, 2, 2, 1, 4, 4, 2, 2, 1, 3, 6, 4, 2, 2, 1, 3, 6, 7, 4, 2, 2, 1, 6, 4, 3, 7, 4, 2, 2, 1, 6, 4, 3, 15, 14, 4, 2, 2
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OFFSET
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1,3
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COMMENTS
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T(n,k) is the order of the permutation p of [1,...,n] defined as follows:
Write F={1,2,3,....,n}.
Place F into a "window" of width k, where k <= n. That is, write out the elements from left to right, up to down, with k elements per line.
Produce a new set F' by traversing the set according to the following algorithm, adding elements to F' as they are traversed in F.
Traversal algorithm:
1) Start at the upper right hand element.
2) If there is an element below the current one
then
A) go to it
B) go back to step 2
3) Otherwise, if there is a column to the left of the current one, then
A) go to it
B) go back to step 2
4) End
Then p is the permutation that sends F to F'.
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LINKS
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Table of n, a(n) for n=1..45.
Samuel Minter, Abulsme function information and definition
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EXAMPLE
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To find T(12,5):
Start with F = { A B C D E F G H I J K L } with a window of widhth 5:
A B C D E
F G H I J
K L
Now let's traverse that and construct our new set
Upper right is E so add it to our new set:
{ E ....
We can go down so we do so and get J
{ E J .....
Now we can't go down so go to the top of the column to the left and get D
{ E J D .....
Eventually we will get:
F' = { E J D I C H B G L A F K }
The permutation p that sends F to F' is a single cycle of length 12, so T(12,5) = 12.
Array begins:
k = 1: 1,1,1,1,1,1,1,1,1,1,... (A000012)
k = 2: 2,2,4,4,3,3,6,6,10,10,... (A024222)
k = 3: 2,2,4,6,6,4,4,4,21,3,... (A118960)
k = 4: 2,2,4,7,3,3,8,10,6,6,... (A120280)
k = 5: 2,2,4,7,15,5,5,12,40,45,... (A120363)
k = 6: 2,2,4,14,6,10,12,12,7,15,... (A120654)
k = 7: 2,2,4,14,6,12,30,4,4,20,... (A121514)
k = 8: 2,2,4,14,6,13,13,24,8,8,...
k = 9: 2,2,4,14,6,13,15,15,63,9,...
k = 10: 2,2,4,14,6,13,16,10,18,12,...
... (Rows converge to A121526)
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PROG
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(C program) int abulsme(int l, int s) {
long int t[30000], m[30000], c[30000], b[30000];
long int k, i, n, j, z, u, q, g;
for (t[1]=s, k=2; k<=l; k++) {
m[k]=(t[k-1]+s-l+abs(t[k-1]+s-l))/(2*abs(t[k-1]+s-l-1)+2);
t[k]=((t[k-1]-m[k])%(s*m[k]+2*l*abs(m[k]-1)))+s*abs(m[k]-1);
}
for (i=1; i<=l; b[i]=0, i++);
for (n=0, i=1; i<=l; i++) {
if (!b[i]) {
j=i;
k=0;
do {
j=t[j];
b[j]=1;
k++;
} while (j!=i);
u=1;
z=1;
if (i>1) {
do {
if (c[z]==k) {
u=0;
}
z++;
} while (!((z>n)||(!u)));
}
if (u) {
n++;
c[n]=k;
}
}
for (q=c[1], g=q, z=1; z<n; z++, g=q) {
for (0; q%c[z+1]; q+=g);
}
}
return g;
}
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CROSSREFS
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Sequence in context: A255716 A177954 A077053 * A060438 A106190 A029290
Adjacent sequences: A105269 A105270 A105271 * A105273 A105274 A105275
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KEYWORD
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nonn,more,tabl
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AUTHOR
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N. J. A. Sloane, Aug 10 2008, based on email from Samuel Minter (abulsme(AT)abulsme.com0, May 08 2008
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STATUS
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approved
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