The chess knight has a unique movement, it may move two squares vertically and one square horizontally, or two squares horizontally and one square vertically (with both forming the shape of an L). The possible movements of chess knight are shown in this diagaram:
A chess knight can move as indicated in the chess diagram below:

We have a chess knight and a phone pad as shown below, the knight can only stand on a numeric cell (i.e. blue cell).

Given an integer n, return how many distinct phone numbers of length n we can dial.
You are allowed to place the knight on any numeric cell initially and then you should perform n - 1 jumps to dial a number of length n. All jumps should be valid knight jumps.
As the answer may be very large, return the answer modulo 109 + 7.
 
Example 1:
Input: n = 1
Output: 10
Explanation: We need to dial a number of length 1, so placing the knight over any numeric cell of the 10 cells is sufficient.

Example 2:
Input: n = 2
Output: 20
Explanation: All the valid number we can dial are [04, 06, 16, 18, 27, 29, 34, 38, 40, 43, 49, 60, 61, 67, 72, 76, 81, 83, 92, 94]

Example 3:
Input: n = 3
Output: 46

Example 4:
Input: n = 4
Output: 104

Example 5:
Input: n = 3131
Output: 136006598
Explanation: Please take care of the mod.

 
Constraints:

1 <= n <= 5000

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'class Solution:\n    def knightDialer(self, n: int) -> int:\n        valid_positions = {\n            (0, 0): 1,\n            (1, 0): 2,\n            (2, 0): 3,\n            (0, 1): 4,\n            (1, 1): 5,\n            (2, 1): 6,\n            (0, 2): 7,\n            (1, 2): 8,\n            (2, 2): 9,\n            (1, 3): 0\n        }\n        \n        mod_val = 10**9 + 7\n        \n        cache = {}\n        def find_numbers(start, length):\n            if start not in valid_positions:\n                return 0\n            x,y = start\n            \n            if (x,y,length) in cache:\n                return cache[(x,y,length)]\n                \n            if length == n:\n                return 1\n            \n            x,y = start\n            nums_found = sum(\n                        [find_numbers((x+1, y-2), length+1),\n                        find_numbers((x+2, y-1), length+1),\n                        find_numbers((x-1, y-2), length+1),\n                        find_numbers((x-2, y-1), length+1),\n                        find_numbers((x-2, y+1), length+1),\n                        find_numbers((x-1, y+2), length+1),\n                        find_numbers((x+2, y+1), length+1),\n                        find_numbers((x+1, y+2), length+1)]\n            ) % mod_val\n            \n            cache[(x,y,length)] = nums_found\n            return nums_found\n                \n                \n            \n        numbers_found = 0\n        for position in valid_positions:\n            numbers_found += find_numbers(position, 1)\n            numbers_found %= mod_val\n            \n        # print(sorted(nums))\n        \n        return numbers_found\n            \n        \n        \n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'knightDialer'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 6790