Padovan n-step number sequences

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.

The Fibonacci-like sequences can be defined like this:

   For n == 2:
       start:      1, 1
       Recurrence: R(n, x) = R(n, x-1) + R(n, x-2); for n == 2
   For n == N:
       start:      First N terms of R(N-1, x)
       Recurrence: R(N, x) = sum(R(N, x-1) + R(N, x-2) + ... R(N, x-N))

For this task we similarly define terms of the first 2..n-step Padowan sequences as:

   For n == 2:
       start:      1, 1, 1
       Recurrence: R(n, x) = R(n, x-2) + R(n, x-3); for n == 2
   For n == N:
       start:      First N + 1 terms of R(N-1, x)
       Recurrence: R(N, x) = sum(R(N, x-2) + R(N, x-3) + ... R(N, x-N-1))

The initial values of the sequences are:

Padovan $n$ -step sequences
$n$	Values	OEIS Entry
2	1,1,1,2,2,3,4,5,7,9,12,16,21,28,37, ...	A134816: 'Padovan's spiral numbers'
3	1,1,1,2,3,4,6,9,13,19,28,41,60,88,129, ...	A000930: 'Narayana's cows sequence'
4	1,1,1,2,3,5,7,11,17,26,40,61,94,144,221, ...	A072465: 'A Fibonacci-like model in which each pair of rabbits dies after the birth of their 4th litter'
5	1,1,1,2,3,5,8,12,19,30,47,74,116,182,286, ...	A060961: 'Number of compositions (ordered partitions) of n into 1's, 3's and 5's'
6	1,1,1,2,3,5,8,13,20,32,51,81,129,205,326, ...	<not found>
7	1,1,1,2,3,5,8,13,21,33,53,85,136,218,349, ...	A117760: 'Expansion of 1/(1 - x - x^3 - x^5 - x^7)'
8	1,1,1,2,3,5,8,13,21,34,54,87,140,225,362, ...	<not found>

Task

Write a function to generate the first $t$ terms, of the first 2..max_n Padovan $n$ -step number sequences as defined above.
Use this to print and show here at least the first t=15 values of the first 2..8 $n$ -step sequences.
(The OEIS column in the table above should be omitted).

Factor

Works with: Factor version 0.99 2021-02-05

<lang factor>USING: compiler.tree.propagation.call-effect io kernel math math.ranges prettyprint sequences ;

padn ( m n -- seq )

   V{ "|" 1 1 1 } over prefix clone over 2 -
   [ dup last2 + suffix! ] times rot pick 1 + -
   [ dup length 1 - pick [ - ] keepd pick <slice> sum suffix! ]
   times nip ;

"Padovan n-step sequences" print 2 8 [a..b] [ 15 swap padn ] map simple-table.</lang>

Output:

Padovan n-step sequences
2 | 1 1 1 2 2 3 4 5  7  9  12 16 21  28  37
3 | 1 1 1 2 3 4 6 9  13 19 28 41 60  88  129
4 | 1 1 1 2 3 5 7 11 17 26 40 61 94  144 221
5 | 1 1 1 2 3 5 8 12 19 30 47 74 116 182 286
6 | 1 1 1 2 3 5 8 13 20 32 51 81 129 205 326
7 | 1 1 1 2 3 5 8 13 21 33 53 85 136 218 349
8 | 1 1 1 2 3 5 8 13 21 34 54 87 140 225 362

Julia

Translation of: Python

<lang julia> """

   First nterms terms of the first 2..max_nstep -step Padowan sequences.

""" function nstep_Padowan(max_nstep=8, nterms=15)

   start = [[], [1, 1, 1]]     # for n=0 and n=1 (hidden).
   for n in 2:max_nstep
       this = start[n][1:n+1]     # Initialise from last
       while length(this) < nterms
           push!(this, sum([this[end - i] for i in 1:n]))
       end
       push!(start, this)
   end
   return start[3:end]

end

function print_Padowan_seq(p)

   println(strip("""

""")) for (n, seq) in enumerate(p) println("| $n || $(replace(string(seq[2:end]), r"[ a-zA-Z\[\]]+" => "")), ...\n|-") end println("|}") end print_Padowan_seq(nstep_Padowan()) </lang>

Output:

Padovan $n$ -step sequences
$n$	Values

Padovan $n$ -step sequences
$n$	Values
1	1,1,2,2,3,4,5,7,9,12,16,21,28,37, ...
2	1,1,2,3,4,6,9,13,19,28,41,60,88,129, ...
3	1,1,2,3,5,7,11,17,26,40,61,94,144,221, ...
4	1,1,2,3,5,8,12,19,30,47,74,116,182,286, ...
5	1,1,2,3,5,8,13,20,32,51,81,129,205,326, ...
6	1,1,2,3,5,8,13,21,33,53,85,136,218,349, ...
7	1,1,2,3,5,8,13,21,34,54,87,140,225,362, ...

Python

Generates a wikitable formatted output <lang python>def pad_like(max_n=8, t=15):

   """
   First t terms of the first 2..max_n-step Padowan sequences.
   """
   start = [[], [1, 1, 1]]     # for n=0 and n=1 (hidden).
   for n in range(2, max_n+1):
       this = start[n-1][:n+1]     # Initialise from last
       while len(this) < t:
           this.append(sum(this[i] for i in range(-2, -n - 2, -1)))
       start.append(this)
   return start[2:]

def pr(p):

   print(

.strip()) for n, seq in enumerate(p, 2): print(f"| {n:2} || {str(seq)[1:-1].replace(' ', )+', ...'}\n|-") print('|}') if __name__ == '__main__': p = pad_like() pr(p)</lang>

Output:

Padovan $n$ -step sequences
$n$	Values

Padovan $n$ -step sequences
$n$	Values
2	1,1,1,2,2,3,4,5,7,9,12,16,21,28,37, ...
3	1,1,1,2,3,4,6,9,13,19,28,41,60,88,129, ...
4	1,1,1,2,3,5,7,11,17,26,40,61,94,144,221, ...
5	1,1,1,2,3,5,8,12,19,30,47,74,116,182,286, ...
6	1,1,1,2,3,5,8,13,20,32,51,81,129,205,326, ...
7	1,1,1,2,3,5,8,13,21,33,53,85,136,218,349, ...
8	1,1,1,2,3,5,8,13,21,34,54,87,140,225,362, ...

Wren

Library: Wren-fmt

<lang ecmascript>import "/fmt" for Fmt

var padovanN // recursive padovanN = Fn.new { |n, t|

   if (n < 2 || t < 3) return [1] * t
   var p = padovanN.call(n-1, t)
   if (n + 1 >= t) return p
   for (i in n+1...t) {
       p[i] = 0
       for (j in i-2..i-n-1) p[i] = p[i] + p[j]
   }
   return p

}

var t = 15 System.print("First %(t) terms of the Padovan n-step number sequences:") for (n in 2..8) Fmt.print("$d: $3d" , n, padovanN.call(n, t))</lang>

Output:

First 15 terms of the Padovan n-step number sequences:
2:   1   1   1   2   2   3   4   5   7   9  12  16  21  28  37
3:   1   1   1   2   3   4   6   9  13  19  28  41  60  88 129
4:   1   1   1   2   3   5   7  11  17  26  40  61  94 144 221
5:   1   1   1   2   3   5   8  12  19  30  47  74 116 182 286
6:   1   1   1   2   3   5   8  13  20  32  51  81 129 205 326
7:   1   1   1   2   3   5   8  13  21  33  53  85 136 218 349
8:   1   1   1   2   3   5   8  13  21  34  54  87 140 225 362