60 lines
3.4 KiB
Text
60 lines
3.4 KiB
Text
These number series are an expansion of the ordinary [[Fibonacci sequence]] where:
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# 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>
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# 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>
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# 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>...
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# 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>
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For small values of <math>n</math>, [[wp:Number prefix#Greek_series|Greek numeric prefixes]] are sometimes used to individually name each series.
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:::: {| style="text-align: left;" border="4" cellpadding="2" cellspacing="2"
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|+ Fibonacci <math>n</math>-step sequences
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|- style="background-color: rgb(255, 204, 255);"
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! <math>n</math> !! Series name !! Values
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|-
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| 2 || fibonacci || 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ...
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|-
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| 3 || tribonacci || 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 ...
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|-
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| 4 || tetranacci || 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 ...
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|-
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| 5 || pentanacci || 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 ...
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|-
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| 6 || hexanacci || 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 ...
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|-
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| 7 || heptanacci || 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 ...
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|-
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| 8 || octonacci || 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 ...
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|-
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| 9 || nonanacci || 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 ...
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|-
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| 10 || decanacci || 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 ...
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|}
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Allied sequences can be generated where the initial values are changed:
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: '''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.
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<!-- Lucas numbers, Lucas number, Lucas series [added to make searches easier.] -->
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<br>
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;Task:
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# 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.
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# Use this to print and show here at least the first ten members of the Fibo/tribo/tetra-nacci and Lucas sequences.
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;Related tasks:
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* [[Fibonacci sequence]]
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* [http://mathworld.wolfram.com/Fibonaccin-StepNumber.html Wolfram Mathworld]
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* [[Hofstadter Q sequence]]
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* [[Leonardo numbers]]
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;Also see:
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* [https://www.youtube.com/watch?v=PeUbRXnbmms Lucas Numbers - Numberphile] (Video)
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* [https://www.youtube.com/watch?v=fMJflV_GUpU Tribonacci Numbers (and the Rauzy Fractal) - Numberphile] (Video)
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* [[wp:Lucas number|Wikipedia, Lucas number]]
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* [http://mathworld.wolfram.com/FibonacciNumber.html MathWorld, Fibonacci Number]
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* [http://www.math-cs.ucmo.edu/~curtisc/articles/howardcooper/genfib4.pdf Some identities for r-Fibonacci numbers]
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* [[oeis:A000045|OEIS Fibonacci numbers]]
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* [[oeis:A000032|OEIS Lucas numbers]]
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<br><br>
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