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| The '''Padovan sequence''' is the [[sequence]] of [[integer]]s ''P''(''n'') defined by the initial values
| | In any given year, about seven percent - between 13 million and 14 million people - will experience a depressive disorder,. This feeling of fear can turn out to be severe and get in the way of accomplishing work, school, and other everyday activities. Run a movie on your mind of the happiest memories you have until you fall back asleep. After all, we learned in school anything that changes the way the mind or body works is a drug. We are constantly looking to the next thing, the better thing, what will happen tomorrow, a week from now, maybe even next year. <br><br>WHY DO YOU IGNORE WARNING SIGNS & HINTS WHICH YOU MUST KNOW BY HEART ALREADY. And these therapy consequences are not dangerous by any means, you're just loosing fat, not loosing electrolytes. If, for example, a child lives in a home where they are exposed to math, they will grow up and most likely acquire good math skills. This disorder often seen in adolescence period but not guaranteed it can occur at any time. Want your child to triumph over social anxiety child disorder; The Anxiety Free. <br><br>This is the most practical way of eliminating any kind of fear and this works well for anxiety attacks. Dogs that are well exercised spend more of their idle time resting rather than panicking. If you have a issue with severe Dachshund anxiety that starts after you leave , you have to initially decrease the time you are away. When this happens, everything becomes more difficult. It may not seem so, but your mind is much more powerful in treating (and bringing on) anxiety than any drug ever could be. <br><br>- Mental 'muddiness' (inability to focus or concentrate). ' Only those couple between the ages of 30 and 50 may adopt kids. When that big ol' bear was breathing down our ancestor's neck, his adrenaline surged as a warning, causing his liver to release energy-stimulating sugars into his system to ready him for the fight. Keeping up with reading is one of the best things any parent can do to keep their child's mind fresh and ready to learn. Instead of thinking the worst will happen, try to change your thought process into something that is positive. <br><br>3) Start attending Social Anxiety Forums that would help you to capture your worst enemies (facing the public). Personally, I don't have an issue with how you decide to cure your anxiety for the most part. A person with the anxiety problem will need to assess what are the priorities in life and then know how to deal with the situation. In that article I explain the best methods for getting your dog acclimated to his crate to the point where he actually looks forward to time in his safe haven. In fact, more often than not, anxiety is not the cause of high blood pressure.<br><br>If you have any queries regarding wherever and how to use anti anxiety diet ([http://www.stress-lavoro-correlato.info/ www.stress-lavoro-correlato.info]), you can speak to us at the internet site. |
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| :<math>P(1)=P(2)=P(3)=1,</math>
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| and the [[recurrence relation]] | |
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| :<math>P(n)=P(n-2)+P(n-3).</math>
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| The first few values of ''P''(''n'') are
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| :1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ... {{OEIS|id=A000931}}
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| [[Image:Padovan triangles (1).png|thumb|350px|Spiral of equilateral triangles with side lengths which follow the Padovan sequence.]]
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| The Padovan sequence is named after [[Richard Padovan]] who attributed its discovery to [[Netherlands|Dutch]] architect [[Hans van der Laan]] in his 1994 essay ''Dom. Hans van der Laan : Modern Primitive''. The sequence was described by [[Ian Stewart (mathematician)|Ian Stewart]] in his [[Scientific American]] column ''Mathematical Recreations'' in June 1996. He also writes about it in one of his books, "Math Hysteria: Fun Games With Mathematics".
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| ''The above definition is the one given by Ian Stewart and by [[MathWorld]]. Other sources may start the sequence at a different place, in which case some of the identities in this article must be adjusted with appropriate offsets.''
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| ==Recurrence relations==
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| In the spiral, each triangle shares a side with two others giving a visual proof that
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| the Padovan sequence also satisfies the recurrence relation | |
| :<math>P(n)=P(n-1)+P(n-5)</math>
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| Starting from this, the defining recurrence and other recurrences as they are discovered,
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| one can create an infinite number of further recurrences by repeatedly replacing <math>P(m)</math> by <math>P(m - 2) + P(m - 3)</math>
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| The [[Perrin pseudoprime|Perrin sequence]] satisfies the same recurrence relations as the Padovan sequence, although it has different initial values. This is a property of recurrence relations.
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| The Perrin sequence can be obtained from the Padovan sequence by the
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| following formula:
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| :<math>\mathrm{Perrin}(n)=P(n+1)+P(n-10).\,</math>
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| ==Extension to negative parameters==
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| As with any sequence defined by a recurrence relation, Padovan numbers ''P''(''m'') for ''m<0'' can be defined by rewriting the recurrence relation as
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| :<math>P(m) = P(m+3) - P(m+1),</math>
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| Starting with ''m=-1'' and working backwards, we extend ''P''(''m'') to negative indices:
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| :{| class="wikitable" style="text-align:right"
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| |-
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| | ''P''<sub>−20</sub>
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| | ''P''<sub>−19</sub>
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| | ''P''<sub>−18</sub>
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| | ''P''<sub>−17</sub>
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| | ''P''<sub>−16</sub>
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| | ''P''<sub>−15</sub>
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| | ''P''<sub>−14</sub>
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| | ''P''<sub>−13</sub>
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| | ''P''<sub>−12</sub>
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| | ''P''<sub>−11</sub>
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| | ''P''<sub>-10</sub>
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| | ''P''<sub>-9</sub>
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| | ''P''<sub>-8</sub>
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| | ''P''<sub>-7</sub>
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| | ''P''<sub>-6</sub>
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| | ''P''<sub>-5</sub>
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| | ''P''<sub>-4</sub>
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| | ''P''<sub>-3</sub>
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| | ''P''<sub>-2</sub>
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| | ''P''<sub>-1</sub>
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| | ''P''<sub>0</sub>
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| | ''P''<sub>1</sub>
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| | ''P''<sub>2</sub>
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| |-
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| | 7
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| | -7
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| | 4
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| | 0
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| | −3
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| | 4
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| | −3
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| | 1
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| | 1
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| | -2
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| | 2
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| | -1
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| | 0
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| | 1
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| | -1
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| | 1
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| | 0
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| | 0
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| | 1
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| | 0
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| | 1
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| | 1
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| | 1
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| |}
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| ==Sums of terms==
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| The sum of the first ''n'' terms in the Padovan sequence is 2 less than ''P''(''n'' + 5) i.e.
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| :<math>\sum_{m=0}^n P(m)=P(n+5)-2.</math>
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| Sums of alternate terms, sums of every third term and sums of every fifth term are also related to other terms in the sequence:
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| :<math>\sum_{m=0}^n P(2m)=P(2n+3)-1</math> {{OEIS2C|A077855}}
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| :<math>\sum_{m=0}^n P(2m+1)=P(2n+4)-1</math>
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| :<math>\sum_{m=0}^n P(3m)=P(3n+2)</math> {{OEIS2C|A034943}}
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| :<math>\sum_{m=0}^n P(3m+1)=P(3n+3)-1</math>
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| :<math>\sum_{m=0}^n P(3m+2)=P(3n+4)-1</math>
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| :<math>\sum_{m=0}^n P(5m)=P(5n+1).</math> {{OEIS2C|A012772}}
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| Sums involving products of terms in the Padovan sequence satisfy the following identities:
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| :<math>\sum_{m=0}^n P(m)^2=P(n+2)^2-P(n-1)^2-P(n-3)^2</math>
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| :<math>\sum_{m=0}^n P(m)^2P(m+1)=P(n)P(n+1)P(n+2)</math>
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| :<math>\sum_{m=0}^n P(m)P(m+2)=P(n+2)P(n+3)-1.</math>
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| ==Other identities==
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| The Padovan sequence also satisfies the identity
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| :<math>P(n)^2-P(n+1)P(n-1)=P(-n-7).\,</math>
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| The Padovan sequence is related to sums of [[binomial coefficient]]s by the following identity:
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| :<math> \sum_{2m+n=k}{m \choose n}=P(k-2).</math>
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| For example, for ''k'' = 12, the values for the pair (''m'', ''n'') with 2''m'' + ''n'' = 12 which give non-zero binomial coefficients are (6, 0), (5, 2) and (4, 4), and:
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| :<math>{6 \choose 0}+{5 \choose 2}+{4 \choose 4}=1+10+1=12=P(10).\,</math>
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| ==Binet-like formula==
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| The Padovan sequence numbers can be written in terms of powers of the roots of the equation
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| :<math> x^3 -x -1 = 0.\,</math>
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| This equation has 3 roots; one real root ''p'' (known as the [[plastic number]]) and two complex conjugate roots ''q'' and ''r''. Given these three roots, the Padovan sequence can be expressed by a formula involving p,q and r:
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| :<math>P\left(n\right) = a p^n + b q^n + c r^n</math>
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| where ''a'', ''b'' and ''c'' are constants.
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| Since the magnitudes of the complex roots ''q'' and ''r'' are both less than 1 (and hence ''p'' is a [[Pisot–Vijayaraghavan number]]), the powers of these roots approach 0 for large ''n'', and <math>P\left(n\right) - a {p^n}</math> tends to zero.
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| <!-- it would be nice to use a p^n below instead of p^(n-1)/s, but it would be nice to express that a as a cubic root as well. Failing that, I prefer the older formula. ~~~~ -->
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| For all <math>n \ge 0</math>, P(n) is the integer closest to <math> \frac {p^{n-1}} {s}</math>,
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| where ''s'' = ''p''/''a'' = 1.0453567932525329623... is the only real root of ''s''<sup>3</sup> − 2''s''<sup>2</sup> + 23''s'' − 23 = 0. The ratio of successive terms in the Padovan sequence approaches ''p'', which has a value of approximately 1.324718. This constant bears the same relationship to the Padovan sequence
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| and the [[Perrin sequence]] as the [[golden ratio]] does to the Fibonacci sequence.
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| ==Combinatorial interpretations==
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| * ''P''(''n'') is the number of ways of writing ''n'' + 2 as an ordered sum in which each term is either 2 or 3 (i.e. the number of [[composition (number theory)|compositions]] of ''n'' + 2 in which each term is either 2 or 3). For example, ''P''(6) = 4, and there are 4 ways to write 8 as an ordered sum of 2s and 3s:
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| ::2 + 2 + 2 + 2 ; 2 + 3 + 3 ; 3 + 2 + 3 ; 3 + 3 + 2
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| * The number of ways of writing ''n'' as an ordered sum in which no term is 2 is ''P''(2''n'' − 2). For example, ''P''(6) = 4, and there are 4 ways to write 4 as an ordered sum in which no term is 2:
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| ::4 ; 1 + 3 ; 3 + 1 ; 1 + 1 + 1 + 1
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| * The number of ways of writing ''n'' as a palindromic ordered sum in which no term is 2 is ''P''(''n''). For example, ''P''(6) = 4, and there are 4 ways to write 6 as a palindromic ordered sum in which no term is 2:
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| ::6 ; 3 + 3 ; 1 + 4 + 1 ; 1 + 1 + 1 + 1 + 1 + 1
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| * The number of ways of writing ''n'' as an ordered sum in which each term is odd and greater than 1 is equal to ''P''(''n'' − 5). For example, ''P''(6) = 4, and there are 4 ways to write 11 as an ordered sum in which each term is odd and greater than 1:
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| ::11 ; 5 + 3 + 3 ; 3 + 5 + 3 ; 3 + 3 + 5
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| * The number of ways of writing ''n'' as an ordered sum in which each term is congruent to 2 mod 3 is equal to ''P''(''n'' − 4). For example, ''P''(6) = 4, and there are 4 ways to write 10 as an ordered sum in which each term is congruent to 2 mod 3:
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| ::8 + 2 ; 2 + 8 ; 5 + 5 ; 2 + 2 + 2 + 2 + 2
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| ==Generating function==
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| The [[generating function]] of the Padovan sequence is
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| :<math>G(P(n);x)=\frac{1+x}{1-x^2-x^3}.</math>
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| This can be used to prove identities involving products of the Padovan sequence with geometric terms, such as:
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| :<math>\sum_{n=0}^{\infty}\frac{P(n)}{2^n} = \frac{12}{5}.</math>
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| :<math>\sum_{n=0}^\infty \frac{P(n)}{\alpha^n} = \frac{\alpha^2(\alpha+1)}{\alpha^3-\alpha-1}.</math>
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| == Generalizations ==
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| In a similar way to the [[Fibonacci numbers]] that can be generalized to a set of polynomials
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| called the [[Fibonacci polynomials]], the Padovan sequence numbers can be generalized to
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| yield the [[Padovan polynomials]].
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| == Padovan prime ==
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| A '''Padovan prime''' is ''P''(''n'') that is [[prime number|prime]]. The first few Padovan primes are | |
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| :2, 3, 5, 7, 37, 151, 3329, 23833, .... {{OEIS|id=A100891}}
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| == Padovan L-system ==
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| If we define the following simple grammar:
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| : '''variables''' : A B C
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| : '''constants''' : none
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| : '''start''' : A
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| : '''rules''' : (A → B), (B → C), (C → AB)
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| then this Lindenmayer system or [[L-system]] produces the following sequence of strings: | |
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| : ''n'' = 0 : A
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| : ''n'' = 1 : B
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| : ''n'' = 2 : C
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| : ''n'' = 3 : AB
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| : ''n'' = 4 : BC
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| : ''n'' = 5 : CAB
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| : ''n'' = 6 : ABBC
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| : ''n'' = 7 : BCCAB
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| : ''n'' = 8 : CABABBC
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| and if we count the length of each string, we obtain the Padovan sequence of numbers:
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| : 1 1 1 2 2 3 4 5 ...
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| Also, if you count the number of ''A''s, ''B''s and ''C''s in each string, then for the ''n''th
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| string, you have ''P''(''n'' − 5) ''A''s, ''P''(''n'' − 3) ''B''s and ''P''(''n'' − 4) ''C''s. The count of ''BB'' pairs, ''AA'' pairs
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| and ''CC'' pairs are also Padovan numbers.
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| == Padovan Cuboid Spiral ==
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| A spiral can be formed based on connecting the corners of a set of 3 dimensional cuboids.
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| This is the [[Padovan cuboid spiral]]. Successive sides of this spiral have lengths that are
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| the Padovan sequence numbers multiplied by the square root of 2.
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| ==External links==
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| *{{SloanesRef |sequencenumber=A000931|name=Padovan sequence}}
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| *{{MathWorld|urlname=PadovanSequence|title=Padovan Sequence}}
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| *[http://www.nexusjournal.com/conferences/N2002-Padovan.html ''Dom Hans Van Der Laan And The Plastic Number''] by Richard Padovan
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| *[http://members.fortunecity.com/templarser/padovan.html ''Tales of a Neglected Number''] by Ian Stewart
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| *[http://www.plenilune.pwp.blueyonder.co.uk/fibonacci-calculator.asp A Padovan Sequence Calculator can be found here.]
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| [[Category:Integer sequences]]
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| [[Category:Recurrence relations]]
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In any given year, about seven percent - between 13 million and 14 million people - will experience a depressive disorder,. This feeling of fear can turn out to be severe and get in the way of accomplishing work, school, and other everyday activities. Run a movie on your mind of the happiest memories you have until you fall back asleep. After all, we learned in school anything that changes the way the mind or body works is a drug. We are constantly looking to the next thing, the better thing, what will happen tomorrow, a week from now, maybe even next year.
WHY DO YOU IGNORE WARNING SIGNS & HINTS WHICH YOU MUST KNOW BY HEART ALREADY. And these therapy consequences are not dangerous by any means, you're just loosing fat, not loosing electrolytes. If, for example, a child lives in a home where they are exposed to math, they will grow up and most likely acquire good math skills. This disorder often seen in adolescence period but not guaranteed it can occur at any time. Want your child to triumph over social anxiety child disorder; The Anxiety Free.
This is the most practical way of eliminating any kind of fear and this works well for anxiety attacks. Dogs that are well exercised spend more of their idle time resting rather than panicking. If you have a issue with severe Dachshund anxiety that starts after you leave , you have to initially decrease the time you are away. When this happens, everything becomes more difficult. It may not seem so, but your mind is much more powerful in treating (and bringing on) anxiety than any drug ever could be.
- Mental 'muddiness' (inability to focus or concentrate). ' Only those couple between the ages of 30 and 50 may adopt kids. When that big ol' bear was breathing down our ancestor's neck, his adrenaline surged as a warning, causing his liver to release energy-stimulating sugars into his system to ready him for the fight. Keeping up with reading is one of the best things any parent can do to keep their child's mind fresh and ready to learn. Instead of thinking the worst will happen, try to change your thought process into something that is positive.
3) Start attending Social Anxiety Forums that would help you to capture your worst enemies (facing the public). Personally, I don't have an issue with how you decide to cure your anxiety for the most part. A person with the anxiety problem will need to assess what are the priorities in life and then know how to deal with the situation. In that article I explain the best methods for getting your dog acclimated to his crate to the point where he actually looks forward to time in his safe haven. In fact, more often than not, anxiety is not the cause of high blood pressure.
If you have any queries regarding wherever and how to use anti anxiety diet (www.stress-lavoro-correlato.info), you can speak to us at the internet site.