Consider a prime number . Let be the -adic valuation.
Let be the sign reduction modulo defined as
if
and
if
. We say
that a triangular numeric pattern with
has triangular symmetry if it is preserved by the dyhedral
group . We show the following facts about binomial
coefficients:

- build a pattern with triangular symmetry for .
- build a pattern with triangular symmetry for .
- is the only composite number such that has triangular symmetry for . The fact that has triangular symmetry was previously observed by A. Granville.
- applied to the last non-zero digit of represented in the number system with base builds a pattern with triangular symmetry for .

Key Words: Binomial coefficient, -adic valuation, triangular symmetry, Kummer's theorem about carries, Pascal's Triangle modulo , automatic -dimensional sequence, Zaphod Beeblebrox.

2010 Mathematics Subject Classification: Primary 11A07; Secondary 05E11, 28A80, 68Q45.

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