A306605 Square array T(n, k) read by antidiagonals upwards, n >= 0 and k >= 0: for any m >= 0, let f_m be the representation of m in the factorial number system: for any i >= 0, 0 <= f_m(i) <= i and m = Sum_{i >= 0} f_m(i) * i!; the representation of T(n, k) in the factorial number system, say g, satisfies g(i) = f_n(f_k(i)) for any i >= 0.

0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 0, 3, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 3, 2, 1, 0, 0, 0, 2, 0, 2, 2, 6, 0, 0, 1, 0, 3, 4, 3, 0, 7, 0, 0, 0, 2, 0, 4, 4, 6, 0, 8, 0, 0, 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, 0, 0, 2, 0, 0, 0, 6, 0, 8, 0, 6, 0, 0, 1, 0, 3, 2, 1, 0, 7, 0

Offset

0,9

Comments

Each column is periodic.

Links

Table of n, a(n) for n=0..86.

Wikipedia, Factorial number system

Index entries for sequences related to factorial base representation

Formula

For any m, n, k >= 0:

- T(n, 0) = T(0, k) = 0 (0 is an absorbing element),

- T(m, T(n, k)) = T(T(m, n), k) (T is associative).

T(n, n) = A306584(n).

Example

Array T(n, k) begins (in decimal):
  n\k|  0  1  2  3  4  5  6  7  8  9  10
  ---+----------------------------------
    0|  0  0  0  0  0  0  0  0  0  0   0
    1|  0  1  2  3  0  1  6  7  8  9   6
    2|  0  0  0  0  2  2  0  0  0  0   2
    3|  0  1  2  3  2  3  6  7  8  9   8
    4|  0  0  0  0  4  4  0  0  0  0   4
    5|  0  1  2  3  4  5  6  7  8  9  10
    6|  0  0  0  0  0  0  0  0  0  0   0
    7|  0  1  2  3  0  1  6  7  8  9   6
    8|  0  0  0  0  2  2  0  0  0  0   2
    9|  0  1  2  3  2  3  6  7  8  9   8
   10|  0  0  0  0  4  4  0  0  0  0   4
Array T(n, k) begins (in factorial base):
   n\k|   0  10  100  110  200  210  1000  1010  1100  1110  1200
  ----+----------------------------------------------------------
     0|   0   0    0    0    0    0     0     0     0     0     0
    10|   0  10  100  110    0   10  1000  1010  1100  1110  1000
   100|   0   0    0    0  100  100     0     0     0     0   100
   110|   0  10  100  110  100  110  1000  1010  1100  1110  1100
   200|   0   0    0    0  200  200     0     0     0     0   200
   210|   0  10  100  110  200  210  1000  1010  1100  1110  1200
  1000|   0   0    0    0    0    0     0     0     0     0     0
  1010|   0  10  100  110    0   10  1000  1010  1100  1110  1000
  1100|   0   0    0    0  100  100     0     0     0     0   100
  1110|   0  10  100  110  100  110  1000  1010  1100  1110  1100
  1200|   0   0    0    0  200  200     0     0     0     0   200

Prog

(PARI) T(n, k) = my (v=0, fn=[]); for (r=1, oo, if (k==0, return (v), fn = concat(fn, n%r); v += fn[1+(k%r)] * (r-1)!; n \= r; k \= r))

Crossrefs

Cf. A007623, A306584 (main diagonal).

Sequence in context: A164615 A182034 A171912 * A363929 A054876 A109502

Adjacent sequences: A306602 A306603 A306604 * A306606 A306607 A306608

Keyword

nonn,base,tabl

Author

Rémy Sigrist, Feb 27 2019

Status

approved