A214679 A(n,k) = Fibonacci(n) represented in bijective base-k numeration; square array A(n,k), n>=1, k>=1, read by antidiagonals.

1, 1, 1, 1, 1, 11, 1, 1, 2, 111, 1, 1, 2, 11, 11111, 1, 1, 2, 3, 21, 11111111, 1, 1, 2, 3, 12, 112, 1111111111111, 1, 1, 2, 3, 11, 22, 221, 111111111111111111111, 1, 1, 2, 3, 5, 14, 111, 1221, 1111111111111111111111111111111111

Offset: 1

Author

Alois P. Heinz, Jul 25 2012

Keywords

• nonn
• tabl

Comments

The digit set for bijective base-k numeration is {1, 2, ..., k}.

Examples

			Square array A(n,k) begins:
:                     1,    1,   1,   1,   1,  1,  1,  1,  1, ...
:                     1,    1,   1,   1,   1,  1,  1,  1,  1, ...
:                    11,    2,   2,   2,   2,  2,  2,  2,  2, ...
:                   111,   11,   3,   3,   3,  3,  3,  3,  3, ...
:                 11111,   21,  12,  11,   5,  5,  5,  5,  5, ...
:              11111111,  112,  22,  14,  13, 12, 11,  8,  8, ...
:         1111111111111,  221, 111,  31,  23, 21, 16, 15, 14, ...
: 111111111111111111111, 1221, 133, 111,  41, 33, 27, 25, 23, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..13
• R. R. Forslund, A logical alternative to the existing positional number system, Southwest Journal of Pure and Applied Mathematics, Vol. 1, 1995, 27-29.
• Eric Weisstein's World of Mathematics, Zerofree
• Wikipedia, Bijective numeration

Crossrefs

Columns k=1-9 give: A108047, A085652, A282234, A282235, A282236, A282237, A282238, A282239, A282240.

Cf. A000045, A214676.

Programs

Maple

with(combinat):
A:= proc(n, b) local d, l, m; m:= fibonacci(n); l:= NULL;
      while m>0 do  d:= irem(m, b, 'm');
        if d=0 then d:=b; m:=m-1 fi;
        l:= d, l
      od; parse(cat(l))
    end:
seq(seq(A(n, 1+d-n), n=1..d), d=1..10);

Mathematica

A[n_, b_] := Module[{d, l, m}, m = Fibonacci@n; l = Nothing; While[m > 0, {m, d} = QuotientRemainder[m, b]; If[d == 0, d = b; m--]; l = {d, l}]; FromDigits @ Flatten @ l];
Table[A[n, d-n+1], {d, 1, 10}, {n, 1, d}] // Flatten (* Jean-François Alcover, May 28 2019, from Maple *)

Formula

A(n,k) = A214676(A000045(n),k).