A214679 - OEIS

%I #28 Jan 08 2021 17:42:46

%S 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,

%T 112,1111111111111,1,1,2,3,11,22,221,111111111111111111111,1,1,2,3,5,

%U 14,111,1221,1111111111111111111111111111111111

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

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

%H Alois P. Heinz, <a href="/A214679/b214679.txt">Antidiagonals n = 1..13</a>

%H R. R. Forslund, <a href="http://www.emis.de/journals/SWJPAM/Vol1_1995/rrf01.ps">A logical alternative to the existing positional number system</a>, Southwest Journal of Pure and Applied Mathematics, Vol. 1, 1995, 27-29.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Zerofree.html">Zerofree</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bijective_numeration">Bijective numeration</a>

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

%e Square array A(n,k) begins:

%e :                     1,    1,   1,   1,   1,  1,  1,  1,  1, ...

%e :                     1,    1,   1,   1,   1,  1,  1,  1,  1, ...

%e :                    11,    2,   2,   2,   2,  2,  2,  2,  2, ...

%e :                   111,   11,   3,   3,   3,  3,  3,  3,  3, ...

%e :                 11111,   21,  12,  11,   5,  5,  5,  5,  5, ...

%e :              11111111,  112,  22,  14,  13, 12, 11,  8,  8, ...

%e :         1111111111111,  221, 111,  31,  23, 21, 16, 15, 14, ...

%e : 111111111111111111111, 1221, 133, 111,  41, 33, 27, 25, 23, ...

%p with(combinat):

%p A:= proc(n, b) local d, l, m; m:= fibonacci(n); l:= NULL;

%p       while m>0 do  d:= irem(m, b, 'm');

%p         if d=0 then d:=b; m:=m-1 fi;

%p         l:= d, l

%p       od; parse(cat(l))

%p     end:

%p seq(seq(A(n, 1+d-n), n=1..d), d=1..10);

%t 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];

%t Table[A[n, d-n+1], {d, 1, 10}, {n, 1, d}] // Flatten (* _Jean-François Alcover_, May 28 2019, from Maple *)

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

%Y Cf. A000045, A214676.

%K nonn,tabl

%O 1,6

%A _Alois P. Heinz_, Jul 25 2012