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 Padovan 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


{\displaystyle n}

-step sequences


 



n


{\displaystyle 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


{\displaystyle t}

 terms, of the first 2..max_n Padovan 



n


{\displaystyle 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


{\displaystyle n}

-step sequences.
 (The OEIS column in the table above should be omitted).


say 'Padovan N-step sequences; first 25 terms:';
 
for 2..8 -> \N {
    my @n-step = 1, 1, 1, { state $n = 2; @n-step[ ($n - N .. $n++ - 1).grep: * >= 0 ].sum } … *;
    put "N = {N} |" ~ @n-step[^25]».fmt: "%5d";
}