48 lines
2.3 KiB
Text
48 lines
2.3 KiB
Text
As the [[Fibonacci sequence]] expands to the [[Fibonacci n-step number sequences]]; We similarly expand the [[Padovan sequence]] to form these Padovan n-step number sequences.
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The Fibonacci-like sequences can be defined like this:
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For n == 2:
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start: 1, 1
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Recurrence: R(n, x) = R(n, x-1) + R(n, x-2); for n == 2
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For n == N:
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start: First N terms of R(N-1, x)
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Recurrence: R(N, x) = sum(R(N, x-1) + R(N, x-2) + ... R(N, x-N))
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For this task we similarly define terms of the first 2..n-step Padovan sequences as:
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For n == 2:
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start: 1, 1, 1
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Recurrence: R(n, x) = R(n, x-2) + R(n, x-3); for n == 2
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For n == N:
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start: First N + 1 terms of R(N-1, x)
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Recurrence: R(N, x) = sum(R(N, x-2) + R(N, x-3) + ... R(N, x-N-1))
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The initial values of the sequences are:
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:: {| style="text-align: left;" border="4" cellpadding="2" cellspacing="2"
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|+ Padovan <math>n</math>-step sequences
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|- style="background-color: rgb(255, 204, 255);"
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! <math>n</math> !! Values !! [https://oeis.org OEIS] Entry
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| 2 || 1,1,1,2,2,3,4,5,7,9,12,16,21,28,37, ... || [https://oeis.org/A134816 A134816]: 'Padovan's spiral numbers'
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| 3 || 1,1,1,2,3,4,6,9,13,19,28,41,60,88,129, ... || [https://oeis.org/A000930 A000930]: 'Narayana's cows sequence'
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| 4 || 1,1,1,2,3,5,7,11,17,26,40,61,94,144,221, ... || [https://oeis.org/A072465 A072465]: 'A Fibonacci-like model in which each pair of rabbits dies after the birth of their 4th litter'
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| 5 || 1,1,1,2,3,5,8,12,19,30,47,74,116,182,286, ... || [https://oeis.org/A060961 A060961]: 'Number of compositions (ordered partitions) of n into 1's, 3's and 5's'
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| 6 || 1,1,1,2,3,5,8,13,20,32,51,81,129,205,326, ... || <not found>
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| 7 || 1,1,1,2,3,5,8,13,21,33,53,85,136,218,349, ... || [https://oeis.org/A117760 A117760]: 'Expansion of 1/(1 - x - x^3 - x^5 - x^7)'
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| 8 || 1,1,1,2,3,5,8,13,21,34,54,87,140,225,362, ... || <not found>
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<br>
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;Task:
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# Write a function to generate the first <math>t</math> terms, of the first <code>2..max_n</code> Padovan <math>n</math>-step number sequences as defined above.
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# Use this to print and show here at least the first <code>t=15</code> values of the first <code>2..8</code> <math>n</math>-step sequences.<br> (The [https://oeis.org OEIS] column in the table above should be omitted).
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<br><br>
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