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

%I A214679 #29 Feb 16 2025 08:33:18
%S A214679 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 A214679 112,1111111111111,1,1,2,3,11,22,221,111111111111111111111,1,1,2,3,5,
%U A214679 14,111,1221,1111111111111111111111111111111111
%N A214679 A(n,k) = Fibonacci(n) represented in bijective base-k numeration; square array A(n,k), n>=1, k>=1, read by antidiagonals.
%C A214679 The digit set for bijective base-k numeration is {1, 2, ..., k}.
%H A214679 Alois P. Heinz, <a href="/A214679/b214679.txt">Antidiagonals n = 1..13</a>
%H A214679 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 A214679 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Zerofree.html">Zerofree</a>
%H A214679 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bijective_numeration">Bijective numeration</a>
%F A214679 A(n,k) = A214676(A000045(n),k).
%e A214679 Square array A(n,k) begins:
%e A214679 :                     1,    1,   1,   1,   1,  1,  1,  1,  1, ...
%e A214679 :                     1,    1,   1,   1,   1,  1,  1,  1,  1, ...
%e A214679 :                    11,    2,   2,   2,   2,  2,  2,  2,  2, ...
%e A214679 :                   111,   11,   3,   3,   3,  3,  3,  3,  3, ...
%e A214679 :                 11111,   21,  12,  11,   5,  5,  5,  5,  5, ...
%e A214679 :              11111111,  112,  22,  14,  13, 12, 11,  8,  8, ...
%e A214679 :         1111111111111,  221, 111,  31,  23, 21, 16, 15, 14, ...
%e A214679 : 111111111111111111111, 1221, 133, 111,  41, 33, 27, 25, 23, ...
%p A214679 with(combinat):
%p A214679 A:= proc(n, b) local d, l, m; m:= fibonacci(n); l:= NULL;
%p A214679       while m>0 do  d:= irem(m, b, 'm');
%p A214679         if d=0 then d:=b; m:=m-1 fi;
%p A214679         l:= d, l
%p A214679       od; parse(cat(l))
%p A214679     end:
%p A214679 seq(seq(A(n, 1+d-n), n=1..d), d=1..10);
%t A214679 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 A214679 Table[A[n, d-n+1], {d, 1, 10}, {n, 1, d}] // Flatten (* _Jean-François Alcover_, May 28 2019, from Maple *)
%Y A214679 Columns k=1-9 give: A108047, A085652, A282234, A282235, A282236, A282237, A282238, A282239, A282240.
%Y A214679 Cf. A000045, A214676.
%K A214679 nonn,tabl
%O A214679 1,6
%A A214679 _Alois P. Heinz_, Jul 25 2012