A108731 Triangle read by rows: row n gives digits of n in factorial base.

0, 1, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 0, 0, 2, 0, 1, 2, 1, 0, 2, 1, 1, 2, 2, 0, 2, 2, 1, 3, 0, 0, 3, 0, 1, 3, 1, 0, 3, 1, 1, 3, 2, 0, 3, 2, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1

Offset

0,7

Comments

Row lengths are A084558. This sequence contains every finite sequence of nonnegative integers.

T(n,k)=A235168(n,k) for k=0..23; a(n) = A235168(n) for n=0..63, when both tables are seen as flattened lists. - Reinhard Zumkeller, Jan 05 2014

Links

Reinhard Zumkeller, Rows n = 0..2000 of triangle, flattened

C. A. Laisant, Sur la numération factorielle, application aux permutations, Bulletin de la Société Mathématique de France, 16 (1888), p. 176-183.

Wikipedia, Factorial number system

Index entries for sequences related to factorial base representation

Example

Triangle begins:
  0
  1
  1, 0
  1, 1
  2, 0
  2, 1
  1, 0, 0
For example, 11 in factorial base is 121 (1*6 + 2*2 + 1*1), so row 11 is 1,2,1.

Maple

b:= proc(n, i) local r; `if`(n<i, [n],
      [b(iquo(n, i, 'r'), i+1)[], r])
    end:
T:= n-> b(n, 2)[]:
seq(T(n), n=0..50);  # Alois P. Heinz, Mar 19 2014

Mathematica

b[n_, i_] := b[n, i] = Module[{q, r}, If[n < i, {n}, {q, r} = QuotientRemainder[n, i]; Append[b[q, i + 1], r]]];
T[n_] := b[n, 2];
Table[T[n], {n, 0, 40}] // Flatten (* Jean-François Alcover, Feb 03 2018, after Alois P. Heinz *)

Prog

(Haskell)
a108731 n k = a108731_row n !! k
a108731_row 0 = [0]
a108731_row n = t n $ reverse $ takeWhile (<= n) $ tail a000142_list
   where t 0 []     = []
         t x (b:bs) = x' : t m bs where (x', m) = divMod x b
a108731_tabf = map a108731_row [0..]
-- Reinhard Zumkeller, Jun 04 2012
(PARI) A108731_row(n)={n=[n]; while(n[1], n=concat(divrem(n[1], 1+#n), n[^1]); n[1]||break); n[^1]~} \\ M. F. Hasler, Jun 20 2017

Crossrefs

Cf: A000142, A007623, A084558, A301872.

Sequence in context: A336167 A076626 A182886 * A235168 A060950 A039976

Adjacent sequences: A108728 A108729 A108730 * A108732 A108733 A108734

Keyword

easy,nonn,tabf,base

Author

Franklin T. Adams-Watters, Jun 22 2005

Extensions

Added a(0)=0 and offset changed accordingly by Reinhard Zumkeller, Jun 04 2012

Status

approved