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Does it conserve all its properties if we consider all systems with a base different from 10?

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    All the intrinsic ones, for sure. The stuff about the digits of $11^k$ for $k<5$ carry on by substituting "$11^k$" with "$11_{b}^k$" and "$k<5$" with "$k<t(b)$", where $t(b)$ is the least integer such that $\binom {t(b)} {\lfloor t(b)/2\rfloor} \ge b$. –  Aug 17 '16 at 07:17
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    Depends what you mean. The fourth row is 1, 4, 6, 4, 1. If you insist on writing the row that way while using base 5, then it's a nonsense, since there's no 6 in base 5. If you mean convert to base 5, so it's 1, 4, 11, 4, 1, then the number in the middle is still six, so it has all the properties six has – unless you're talking about properties like the sum of all the "digits". In short, your question is highly ambiguous, and no answer is possible until you disambiguate. – Gerry Myerson Aug 17 '16 at 07:25
  • @GerryMyerson, sure, of course the numbers must be adapted to the relative base to make the correspondent powers of 11 –  Aug 17 '16 at 07:45
  • A cousin question, may be a little more sound, is : "What is Pascal's triangle looking like when taken modulo n, for different $n$" ? (I agree that is as considering the last digit in a base $n$ representation). A particularly nice case is Pascal's triangle mod $2$ which gives a discrete analog of Sierpinski triangle (https://oeis.org/wiki/Template:Sierpinski's_triangle_(Pascal's_triangle_mod_2)) – Jean Marie Aug 17 '16 at 10:28

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The entries in Pascal's triangle are just numbers -- they are not numbers are in any particular base.

You can choose to write those numbers down in base ten, or base seventeen, or in Roman numerals or English words. That does not change which numbers they are, and every property of the triangle that only depends on which numbers are in it will work the same no matter how you choose to write down those numbers.

Purely typographical "properties" that speak of your representation of numbers rather than what the numbers themselves are -- such as "how many digits there are in row 17" -- will of course change with your choice of representation.

                          eins
                     eins      eins
                eins      zwei      eins
           eins      drei      drei      eins
      eins      vier      sechs     vier      eins
 eins      fünf      zehn      zehn      fünf      eins
hmakholm left over Monica
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  • @user104372: I don't know what you mean by "each row is a power of 11". The numbers are the same no matter which base you write it down in. If you find a Fibonacci number somewhere in the triangle, that Fibonacci number will be at the same position in the triangle no matter which method you use for writing the numbers in the triangle. – hmakholm left over Monica Aug 17 '16 at 07:43