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

Author

Franklin T. Adams-Watters, Jun 22 2005

Keywords

• easy
• nonn
• tabf
• base

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

Examples

			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.
		

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

Crossrefs

Cf: A000142, A007623, A084558, A301872.

Programs

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

Maple

b:= proc(n, i) local r; `if`(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 *)

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

Python

def row(n, i=2): return [n] if n < i else [*row(n//i, i=i+1), n%i]
print([d for n in range(35) for d in row(n)]) # Michael S. Branicky, Oct 02 2024

Extensions

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