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Copy file name to clipboardExpand all lines: docs/src/developerguide.md
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@@ -77,11 +77,11 @@ in a final step.
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All terms of all sequences have the same type. Currently this is the
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type fmpz as provided by the Nemo library.
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Sequences supports the use of notation using unicode characters, especially the
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traditional notation used in number theory. For example we define
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IntegerSequences supports the use of notation using unicode characters, especially the traditional notation used in number theory. For example we define
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```
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τ(n) = Nemo.sigma(n, 0)
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μ(n) = Nemo.moebiusmu(n)
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V006171(n) = EulerTransform(τ)(n)
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```
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We also support new notations like the proposal from Knuth, Graham and
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Patashnik in Concrete Mathematics:
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For example ``⊥(n, ϕ(n))`` indicates if there is just one group of order ``n``.
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But this is not only a concise mathematical formula, this is also valid Julia
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code (defined in Sequences). The predicate gives rise to the sequence of cyclic
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numbers, A003277 in the OEIS.
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code (defined in IntegerSequences). The predicate gives rise to the sequence of cyclic numbers, A003277 in the OEIS.
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Similarly possible definitions of some sequences (not necessarily efficient ones
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in the computational sense) are
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In the OEIS, on the other hand, a sequence is an enumeration, a set with an index function where the first index (called offset o) is specified. With this we arrive at this picture:
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``a_o, b_{o+1}, c_{o+2}, d_{o+3}, ...``
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``a_o, a_{o+1}, a_{o+2}, a_{o+3}, ...``
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In this view a list (representing the initial segment of the sequence)
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takes the place of the iteration.
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``[ a_{o}, b_{o+1}, c_{o+2}, ..., z_{o+n-1} ]``
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``[ a_{o}, a_{o+1}, a_{o+2}, ..., a_{o+n-1} ]``
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In contrast in our setup the concept of offset and indexing does not occur at all but is transferred to the interpretation: only the application decides about indexing and offset. In practice our setup avoiding the use of an offset turns out
Copy file name to clipboardExpand all lines: docs/src/useofoeis.md
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@@ -4,7 +4,7 @@ We emphasize that references to the Online Encyclopedia of Integer Sequences alw
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There is a variety of reasons to deviate occasionally from the definitions used in the OEIS.
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* Many sequences there are frozen by the editor in chief for so-called 'historical reasons' (for example because they were published in this form in printed versions of the encyclopedia).
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* Many sequences there are frozen by the editor in chief for so-called 'historical reasons' (for example because they were published in this form in printed versions of the encyclopedia or referenced in scientific papers).
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* Conventions in the OEIS are used that made more sense in times of punch cards than today, for example the suppression of 0's in many series expansions.
Copy file name to clipboardExpand all lines: src/BellNumbers.jl
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Let ``F`` be an integer sequence generating function, then ``B_{n,k}(F) = \\sum_{m=1}^{n-k+1} \\binom{n-1}{m-1} F(m) B_{n-m,k-1}(F)`` where ``B_{0,0} = 1, B_{n,0} = 0`` for ``n≥1, B_{0,k} = 0`` for ``k≥1``.
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