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# Enanacci numbers

(Redirected from 1-bonacci numbers)

The enanacci numbers (the prefix ένα being 'one' in Greek) (1-bonacci numbers) are degenerate N-bonacci numbers. (Cf. Category:Recurrence, linear, order 01, (1)).

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 ${\displaystyle \scriptstyle N\,=\,1\,}$, we thus have ${\displaystyle \scriptstyle a(0)\,=\,1;\;a(n)\,=\,a(n-1),\,n\,\geq \,1.}$ This begets the all 1's sequence.

A000012 The simplest sequence of positive numbers: the all 1's sequence.

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}