Fibonacci n-step number sequences

Fibonacci n-step number sequences is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

These number series are an expansion of the ordinary Fibonacci sequence where:

1. For n = 2 we have the Fibonacci sequence; with initial values [1, 1] and $F_{k}^{2}=F_{k-1}^{2}+F_{k-2}^{2}$
2. For n = 3 we have the tribonacci sequence; with initial values [1, 1, 2] and $F_{k}^{3}=F_{k-1}^{3}+F_{k-2}^{3}+F_{k-3}^{3}$
3. For n = 4 we have the tetranacci sequence; with initial values [1, 1, 2, 4] and $F_{k}^{4}=F_{k-1}^{4}+F_{k-2}^{4}+F_{k-3}^{4}+F_{k-4}^{4}$
   ...
4. For general n>2 we have the Fibonacci n-step sequence - $F_{k}^{n}$; with initial values of the first n values of the (n-1)'th Fibonacci n-step sequence $F_{k}^{n-1}$; and k'th value of this n'th sequence being $F_{k}^{n}=\sum _{i=1}^{(n)}{F_{k-i}^{(n)}}$

Allied sequences can be generated where the initial values are changed. The Lucas series sums the two preceeding values like the fibonacci series for n==2 but has the values [2, 1] as its initial values.

The task is to:

1. 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.
2. Use this to print and show here the first ten members of the Fibo/tribo/tetra-nacci and Lucas sequences.

C.f

• Fibonacci sequence

Python

<lang python>>>> def fiblike(start): addnum = len(start) def fibber(n): try: return fibber.memo[n] except: 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] >>> f3= fiblike([1,1,2]) # tribo- >>> [f3(i) for i in range(10)] [1, 1, 2, 4, 7, 13, 24, 44, 81, 149] >>> f4 = fiblike([1,1,2,4]) # tetra- >>> [f4(i) for i in range(10)] [1, 1, 2, 4, 8, 15, 29, 56, 108, 208] >>> f5 = fiblike([1,1,2,4,8]) # penta- >>> [f5(i) for i in range(10)] [1, 1, 2, 4, 8, 16, 31, 61, 120, 236] >>> f6 = fiblike([1,1,2,4,8,16]) # hexa- >>> [f6(i) for i in range(10)] [1, 1, 2, 4, 8, 16, 32, 63, 125, 248] >>> f7 = fiblike([1,1,2,4,8,16,32]) # hepta >>> [f7(i) for i in range(10)] [1, 1, 2, 4, 8, 16, 32, 64, 127, 253] >>> </lang>

Tcl

Works with: Tcl version 8.6

<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, ...]