user
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 {\displaystyle n}}
        -step sequences

        | n {\displaystyle {\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

1. Write a function to generate the first 

   t
   {\displaystyle {\displaystyle t}}
    terms, of the first `2..max_n` Padovan 

   n
   {\displaystyle {\displaystyle 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
   {\displaystyle {\displaystyle n}}
   -step sequences.  
    (The OEIS column in the table above should be omitted).
assistant
```
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";
}

```

Output:

```
Padovan N-step sequences; first 25 terms:
N = 2 |    1     1     1     2     2     3     4     5     7     9    12    16    21    28    37    49    65    86   114   151   200   265   351   465   616
N = 3 |    1     1     1     2     3     4     6     9    13    19    28    41    60    88   129   189   277   406   595   872  1278  1873  2745  4023  5896
N = 4 |    1     1     1     2     3     5     7    11    17    26    40    61    94   144   221   339   520   798  1224  1878  2881  4420  6781 10403 15960
N = 5 |    1     1     1     2     3     5     8    12    19    30    47    74   116   182   286   449   705  1107  1738  2729  4285  6728 10564 16587 26044
N = 6 |    1     1     1     2     3     5     8    13    20    32    51    81   129   205   326   518   824  1310  2083  3312  5266  8373 13313 21168 33657
N = 7 |    1     1     1     2     3     5     8    13    21    33    53    85   136   218   349   559   895  1433  2295  3675  5885  9424 15091 24166 38698
N = 8 |    1     1     1     2     3     5     8    13    21    34    54    87   140   225   362   582   936  1505  2420  3891  6257 10061 16178 26014 41830
```