A208279 Central terms of Pascal's triangle mod 10 (A008975).

1, 2, 6, 0, 0, 2, 4, 2, 0, 0, 6, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 4, 8, 4, 0, 0, 2, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 2, 6, 0, 0, 2, 4, 2, 0, 0, 6, 2, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Reinhard Zumkeller, Feb 25 2012

Keywords

• nonn
• base

Comments

From Chai Wah Wu, Dec 08 2023: (Start)

Last digit of central binomial coefficient binomial(2n,n) in base 10.

A000984 mod 10.

A073095 are numbers n such that a(n) = A008904(n).

a(n) is even for n>0. (End)

From Robert Israel, Dec 08 2023: (Start)

If at least one base-5 digit of n is 3 or 4, then a(n) = 0.

Otherwise, if k 1's occur in the base-5 expansion of n, then a(n) = 2^k (mod 10) if k > 0, or 6 if k = 0. (End)

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000984, A008904, A008975, A073095.

Programs

Haskell

a208279 n = a008975 (2*n) n

Maple

f:= proc(n) local A,v;
   A:= convert(n,base,5);
   if select(`>=`,A,3) <> [] then return 0 fi;
   v:= numboccur(1,A);
   if v > 0 then 2^v mod 10
   else 6
   fi
end proc:
f(0):= 1:
map(f, [$0..200]); # Robert Israel, Dec 08 2023

Mathematica

Array[Mod[Binomial[2#,#],10]&,100,0] (* Paolo Xausa, Dec 09 2023 *)

Python

from sympy.ntheory.factor_ import digits
def A208279(n):
    if n == 0: return 1
    s = digits(n,5)[1:]
    return 0 if any(x>2 for x in s) else ((6,2,4,8)[a&3] if (a:=s.count(1)) else 6) # Chai Wah Wu, Dec 08 2023

Formula

a(n) = A008975(2*n,n) = binomial(2n,n) mod 10.