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

1,6

Comments

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

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

Formula

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

Example

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, ...

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 *)

Crossrefs

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

Cf. A000045, A214676.

Sequence in context: A145140 A010195 A010193 * A332499 A010192 A214326

Adjacent sequences: A214676 A214677 A214678 * A214680 A214681 A214682

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 25 2012

Status

approved