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 A105272 Array T(n,k) (k >= 1, n >= k) read by antidiagonals (see definition in Comments lines). 8
 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, 1, 10, 4, 8, 5, 6, 14, 4, 2, 2, 1, 10, 21, 10, 5, 10, 6, 14, 4, 2, 2, 1, 12, 3, 6, 12, 12, 12, 6, 14, 4, 2, 2 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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'. LINKS Robert Price, Table of n, a(n) for n = 1..1275 Samuel Minter, Abulsme function information and definition EXAMPLE 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) MATHEMATICA T[1] = ConstantArray[1, 75]; For[k = 2, k <= 20, k++, T[k] = Table[f = Range[n]; fp = {}; For[col = k, col > 0, col--, For[row = 0, col + row*k <= n, row++, AppendTo[fp, f[[col + row*k]]]]]; LCM @@ Length /@ First[FindPermutation[f, fp]], {n, k, 75}]]; A105272 = {}; For[i = 1, i <= 20, i++, For[j = i, j >= 1, j--, AppendTo[A105272, T[i - j + 1][[j]]]]]; A105272 (* Robert Price, Aug 26 2019 *) PROG (C) 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; } CROSSREFS Sequence in context: A363345 A344651 A308558 * A060438 A106190 A029290 Adjacent sequences: A105269 A105270 A105271 * A105273 A105274 A105275 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, Aug 10 2008, based on email from Samuel Minter (abulsme(AT)abulsme.com0, May 08 2008 EXTENSIONS a(46)-a(78) from Robert Price, Aug 26 2019 STATUS approved

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Last modified August 13 20:30 EDT 2024. Contains 375144 sequences. (Running on oeis4.)