Fibonacci n-step number sequences: Difference between revisions
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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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| 8 || octonacci || 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 ... |
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| 9 || nonanacci || 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 ... |
| 9 || nonanacci || 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 ... |
Revision as of 06:10, 25 May 2012
These number series are an expansion of the ordinary Fibonacci sequence where:
- For we have the Fibonacci sequence; with initial values and
- For we have the tribonacci sequence; with initial values and
- For we have the tetranacci sequence; with initial values and
... - For general we have the Fibonacci n-step sequence - ; with initial values of the first n values of the (n-1)'th Fibonacci n-step sequence ; and k'th value of this n'th sequence being
For small values of n, Greek numeric prefixes are sometimes used to individually name each series.
Fibonacci n-step sequences n 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 Lucas series sums the two preceeding values like the fibonacci series for but has the values as its initial values.
- The task is to
- Write a function to generate Fibonacci n-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.
- C.f
Python
<lang python>>>> def fiblike(start): addnum = len(start) def fibber(n): try: return fibber.memo[n] except IndexError: ans = sum(fibber(i) for i in range(n-addnum, n)) fibber.memo.append(ans) return ans fibber.memo = start[:] return fibber
>>> fibo = fiblike([1,1]) >>> [fibo(i) for i in range(10)] [1, 1, 2, 3, 5, 8, 13, 21, 34, 55] >>> lucas = fiblike([2,1]) >>> [lucas(i) for i in range(10)] [2, 1, 3, 4, 7, 11, 18, 29, 47, 76] >>> for n, name in zip(range(2,11), 'fibo tribo tetra penta hexa hepta octo nona deca'.split()) : fibber = fiblike([1] + [2**i for i in range(n-1)]) print('n=%2i, %5snacci -> %s ...' % (n, name, ' '.join(str(fibber(i)) for i in range(15))))
n= 2, fibonacci -> 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 ...
n= 3, tribonacci -> 1 1 2 4 7 13 24 44 81 149 274 504 927 1705 3136 ...
n= 4, tetranacci -> 1 1 2 4 8 15 29 56 108 208 401 773 1490 2872 5536 ...
n= 5, pentanacci -> 1 1 2 4 8 16 31 61 120 236 464 912 1793 3525 6930 ...
n= 6, hexanacci -> 1 1 2 4 8 16 32 63 125 248 492 976 1936 3840 7617 ...
n= 7, heptanacci -> 1 1 2 4 8 16 32 64 127 253 504 1004 2000 3984 7936 ...
n= 8, octonacci -> 1 1 2 4 8 16 32 64 128 255 509 1016 2028 4048 8080 ...
n= 9, nonanacci -> 1 1 2 4 8 16 32 64 128 256 511 1021 2040 4076 8144 ...
n=10, decanacci -> 1 1 2 4 8 16 32 64 128 256 512 1023 2045 4088 8172 ...
>>> </lang>
Tcl
<lang tcl>package require Tcl 8.6
proc fibber {args} {
coroutine fib[incr ::fibs]=[join $args ","] apply {fn {
set n [info coroutine] foreach f $fn { if {![yield $n]} return set n $f } while {[yield $n]} { set fn [linsert [lreplace $fn 0 0] end [set n [+ {*}$fn]]] }
} ::tcl::mathop} $args
}
proc print10 cr {
for {set i 1} {$i <= 10} {incr i} {
lappend out [$cr true]
} puts \[[join [lappend out ...] ", "]\] $cr false
} puts "FIBONACCI" print10 [fibber 1 1] puts "TRIBONACCI" print10 [fibber 1 1 2] puts "TETRANACCI" print10 [fibber 1 1 2 4] puts "LUCAS" print10 [fibber 2 1]</lang>
- Output:
FIBONACCI [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...] TRIBONACCI [1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ...] TETRANACCI [1, 1, 2, 4, 8, 15, 29, 56, 108, 208, ...] LUCAS [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, ...]