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# 0-bonacci numbers

The medenacci numbers (the prefix μηδέν being 'nothing' in Greek, though the ancient Greeks had no concept of zero as a number) (0-bonacci numbers) are degenerate N-bonacci numbers.

Abiding by the following definition of N-bonacci numbers

The N-bonacci numbers arise from a recurrence relation like that of the Fibonacci numbers, but with ${\displaystyle \scriptstyle N}$ initial terms defined as

${\displaystyle {\begin{array}{rcl}a(n)&:=&0,\quad 0\leq n\leq N-2;\\a(N-1)&:=&1;\end{array}}}$

instead of two initial terms, and where each subsequent term is the sum of the previous ${\displaystyle \scriptstyle N}$ terms

${\displaystyle a(n):=\sum _{i=n-N}^{n-1}a(i),\quad n\geq N.}$

With ${\displaystyle \scriptstyle N\,=\,0\,}$, we thus have no initial term, then sum of zero previous terms (the upper bound of the summation being lower than the lower bound), giving the empty sum, i.e. 0. This begets the all 0's sequence.

A000004 The zero sequence.

{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}