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Recamán's sequence: 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, ... "subtract if possible, otherwise add": a(0) = 0; for n > 0, a(n) = a(n − 1) − n if that number is positive and not already in the sequence, otherwise a(n) = a(n − 1) + n, whether or not that number is already in the sequence. A005132: Look-and ...
Python sets are very much like mathematical sets, and support operations like set intersection and union. Python also features a frozenset class for immutable sets, see Collection types. Dictionaries (class dict) are mutable mappings tying keys and corresponding values. Python has special syntax to create dictionaries ({key: value})
Since 7 October 2024, Python 3.13 is the latest stable release, and it and, for few more months, 3.12 are the only releases with active support including for bug fixes (as opposed to just for security) and Python 3.9, [55] is the oldest supported version of Python (albeit in the 'security support' phase), due to Python 3.8 reaching end-of-life.
An integer sequence is computable if there exists an algorithm that, given n, calculates a n, for all n > 0. The set of computable integer sequences is countable.The set of all integer sequences is uncountable (with cardinality equal to that of the continuum), and so not all integer sequences are computable.
In terms of mathematical optimization, dynamic programming usually refers to simplifying a decision by breaking it down into a sequence of decision steps over time. This is done by defining a sequence of value functions V 1, V 2, ..., V n taking y as an argument representing the state of the system at times i from 1 to n.
As there are only a finite number of possible n-tuples with these constraints, the sequence of n-tuples must sooner or later repeat itself. Every Ducci sequence therefore eventually becomes periodic. If n is a power of 2 every Ducci sequence eventually reaches the n-tuple (0,0,...,0) in a finite number of steps. [1] [3] [4]
The ordinary generating function of a sequence can be expressed as a rational function (the ratio of two finite-degree polynomials) if and only if the sequence is a linear recursive sequence with constant coefficients; this generalizes the examples above. Conversely, every sequence generated by a fraction of polynomials satisfies a linear ...
A Fibonacci sequence of order n is an integer sequence in which each sequence element is the sum of the previous elements (with the exception of the first elements in the sequence). The usual Fibonacci numbers are a Fibonacci sequence of order 2.