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

1. Write a function to generate the first $t$ terms, of the first 2..max_n Padovan $n$-step number sequences as defined above.
2. 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, ...