These number series are an expansion of the ordinary [[Fibonacci sequence]] where:
# For <math>n = 2</math> we have the Fibonacci sequence; with initial values <math>[1, 1]</math> and <math>F_k^2 = F_{k-1}^2 + F_{k-2}^2</math>
# For <math>n = 3</math> we have the tribonacci sequence; with initial values <math>[1, 1, 2]</math> and <math>F_k^3 = F_{k-1}^3 + F_{k-2}^3 + F_{k-3}^3</math>
# For <math>n = 4</math> we have the tetranacci sequence; with initial values <math>[1, 1, 2, 4]</math> and <math>F_k^4 = F_{k-1}^4 + F_{k-2}^4 + F_{k-3}^4 + F_{k-4}^4</math><br>...
# For general <math>n>2</math> we have the Fibonacci <math>n</math>-step sequence - <math>F_k^n</math>; with initial values of the first <math>n</math> values of the <math>(n-1)</math>'th Fibonacci <math>n</math>-step sequence <math>F_k^{n-1}</math>; and <math>k</math>'th value of this <math>n</math>'th sequence being <math>F_k^n = \sum_{i=1}^{(n)} {F_{k-i}^{(n)}}</math>

For small values of <math>n</math>, [[wp:Number prefix#Greek_series|Greek numeric prefixes]] are sometimes used to individually name each series.

:::: {| style="text-align: left;" border="4" cellpadding="2" cellspacing="2"
|+ Fibonacci <math>n</math>-step sequences
|- style="background-color: rgb(255, 204, 255);"
! <math>n</math> !! Series name !! Values
|-
|  2 ||  fibonacci || 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ...
|-
|  3 || tribonacci || 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 ...
|-
|  4 || tetranacci || 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 ...
|-
|  5 || pentanacci || 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 ...
|-
|  6 ||  hexanacci || 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 ...
|-
|  7 || heptanacci || 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 ...
|-
|  8 ||  octonacci || 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 ...
|-
|  9 ||  nonanacci || 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 ...
|-
| 10 ||  decanacci || 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 ...
|}

Allied sequences can be generated where the initial values are changed:
: '''The [[wp:Lucas number|Lucas series]]''' sums the two preceding values like the fibonacci series for <math>n=2</math> but uses <math>[2, 1]</math> as its initial values.

<!-- Lucas numbers, Lucas number, Lucas series     [added to make searches easier.] -->

<br>
;Task:
# Write a function to generate Fibonacci <math>n</math>-step number sequences given its initial values and assuming the number of initial values determines how many previous values are summed to make the next number of the series.
# Use this to print and show here at least the first ten members of the Fibo/tribo/tetra-nacci and Lucas sequences.


;Related tasks:
* &nbsp; [[Fibonacci sequence]]
* &nbsp; [http://mathworld.wolfram.com/Fibonaccin-StepNumber.html Wolfram Mathworld]
* &nbsp; [[Hofstadter Q sequence‎]]
* &nbsp; [[Leonardo numbers]]


;Also see:
* &nbsp; [https://www.youtube.com/watch?v=PeUbRXnbmms Lucas Numbers - Numberphile] (Video)
* &nbsp; [https://www.youtube.com/watch?v=fMJflV_GUpU Tribonacci Numbers (and the Rauzy Fractal) - Numberphile] (Video)
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