## Task

You have to write three functions namely - `PNum, GPNum and SPNum` (JS, Coffee), `p_num, g_p_num and s_p_num` (Python and Ruby), `pNum, gpNum and spNum` (Java, C#), `p-num, gp-num and sp-num` (Clojure) - to check whether a given argument `n` is a Pentagonal, Generalized Pentagonal, or Square Pentagonal Number, and return `true` if it is and `false` otherwise.

### Description




`Pentagonal Numbers` - The nth pentagonal number Pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots (means the side contains n number of dots), when the pentagons are overlaid so that they share one corner vertex.


> First few Pentagonal Numbers are: 1, 5, 12, 22...

`Generalized Pentagonal Numbers` - All the Pentagonal Numbers along with the number of dots inside the outlines of all the pentagons of a pattern forming a pentagonal number pentagon are called Generalized Pentagonal Numbers.

> First few Generalized Pentagonal Numbers are: 0, 1, 2, 5, 7, 12, 15, 22...

`Square Pentagonal Numbers` - The numbers which are Pentagonal Numbers and are also a perfect square are called Square Pentagonal Numbers. 

> First few are: 1, 9801, 94109401...
 
### Examples

#### Note: 
* Pn = Nth Pentagonal Number
* Gpn = Nth Generalized Pentagonal Number



    
     ^        ^          ^             ^                 ^
    P1=1     P2=5      P3=12         P4=22             P5=35   //Total number of distinct dots used in the Pattern
    Gp2=1    Gp4=5     Gp6=12        Gp8=22                    //All the Pentagonal Numbers are Generalised
             Gp1=0     Gp3=2         Gp5=7             Gp7=15  //Total Number of dots inside the outermost Pentagon

```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 = "def isqrt(num):\n    '''Compute int(sqrt(n)) for n integer > 0\n    O(log4(n)) and no floating point operation, no division'''\n    res, bit = 0, 1\n    while bit <= num:\n        bit <<= 2\n    bit >>= 2\n\n    while bit:\n        if num >= res + bit:\n            num -= res + bit\n            res += bit << 1\n        res >>= 1\n        bit >>= 2\n    return res\n\ndef sqrt(n):\n    # return n ** .5                                 ## Nope\n    # return __import__('math').sqrt(n)              ## Not enough precision...\n    # return __import__('decimal').Decimal(n).sqrt() ## That one works great!\n    return isqrt(n) ## The very best, you don't need those nasty floating point numbers\n\ndef s_num(n):\n    return sqrt(n) ** 2 == n\n\ndef g_p_num(n):\n    return s_num(24 * n + 1)\n\ndef p_num(n):\n    return g_p_num(n) and sqrt(24 * n + 1) % 6 == 5\n\ndef s_p_num(n):\n    return s_num(n) and p_num(n) "
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'p_num'
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: 5696