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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.
10
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
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
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 25 2012
STATUS
approved