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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. 1

%I

%S 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,

%T 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,

%U 2,0,0,0,6,0,8,0,6,0,0,1,0,3,2,1,0,7,0

%N 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.

%C Each column is periodic.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Factorial_number_system">Factorial number system</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

%F For any m, n, k >= 0:

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

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

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

%e Array T(n, k) begins (in decimal):

%e n\k| 0 1 2 3 4 5 6 7 8 9 10

%e ---+----------------------------------

%e 0| 0 0 0 0 0 0 0 0 0 0 0

%e 1| 0 1 2 3 0 1 6 7 8 9 6

%e 2| 0 0 0 0 2 2 0 0 0 0 2

%e 3| 0 1 2 3 2 3 6 7 8 9 8

%e 4| 0 0 0 0 4 4 0 0 0 0 4

%e 5| 0 1 2 3 4 5 6 7 8 9 10

%e 6| 0 0 0 0 0 0 0 0 0 0 0

%e 7| 0 1 2 3 0 1 6 7 8 9 6

%e 8| 0 0 0 0 2 2 0 0 0 0 2

%e 9| 0 1 2 3 2 3 6 7 8 9 8

%e 10| 0 0 0 0 4 4 0 0 0 0 4

%e Array T(n, k) begins (in factorial base):

%e n\k| 0 10 100 110 200 210 1000 1010 1100 1110 1200

%e ----+----------------------------------------------------------

%e 0| 0 0 0 0 0 0 0 0 0 0 0

%e 10| 0 10 100 110 0 10 1000 1010 1100 1110 1000

%e 100| 0 0 0 0 100 100 0 0 0 0 100

%e 110| 0 10 100 110 100 110 1000 1010 1100 1110 1100

%e 200| 0 0 0 0 200 200 0 0 0 0 200

%e 210| 0 10 100 110 200 210 1000 1010 1100 1110 1200

%e 1000| 0 0 0 0 0 0 0 0 0 0 0

%e 1010| 0 10 100 110 0 10 1000 1010 1100 1110 1000

%e 1100| 0 0 0 0 100 100 0 0 0 0 100

%e 1110| 0 10 100 110 100 110 1000 1010 1100 1110 1100

%e 1200| 0 0 0 0 200 200 0 0 0 0 200

%o (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))

%Y Cf. A007623, A306584 (main diagonal).

%K nonn,base,tabl

%O 0,9

%A _Rémy Sigrist_, Feb 27 2019

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Last modified November 19 06:03 EST 2019. Contains 329310 sequences. (Running on oeis4.)