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| In [[mathematics]], an [[integer sequence]] is called a '''complete sequence''' if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.
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| For example, the sequence of [[powers of two]] {1, 2, 4, 8, ...}, based on the [[binary numeral system]], is a complete sequence; given any natural number, we can choose the values corresponding to the 1 bits in its binary representation and sum them to obtain that number (e.g. 37 = 100101<sub>2</sub> = 1 + 4 + 32). This sequence is minimal, since no value can be removed from it without making some natural numbers impossible to represent. Simple examples of sequences that are not complete include:
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| * The [[even number]]s; since adding even numbers produces only even numbers, no odd number can be formed.
| | They have powerful and durable mountain bikes with a high-tech suspension system. Macaskill is tackling the Cuillin Ridge in Scotland, not far from where this two-wheeling genius hails from on the Isle of Skye. These bikes are very popular with the sport of mountain biking. Alison Addy is the author of many articles on subjects like mountain bike crashes and published at. With this in mind all you have to do is book the vacation and find the right bike for you and your holiday. <br><br>Adjust the tyre pressure too so it is at its optimum level. General Use (Commuting, Mountain Bike, Road Bike, BMX, and Universal Fit) General use helmets are usually about. A wider tire will give you more traction when riding over rocks or in turns, you can run at lower pressure and the more cushioned the ride will feel, but they are heavier. Now please continue on additionally so you can get added info for this topic. A sports drink is better than water as they contain carbohydrates and electrolytes that speed absorption of fluids and can give you an energy boost. <br><br>Mountain bikes in Arizona have two great rides in Sedona, Highline and the Dry Creek Loop. * Numbness - Impingement of small nerve branches between the second and third or third and fourth toes can cause swelling that results in numbness, tingling, or burning, or sharp shooting pains into the toes. The whole set-up, including your bag, should weigh no more than 20kgs (44lbs). Once you have answered the above questions and you see all these features present in a bike, you are on your way to choosing the road bike that is right for you. One of the obvious things to check on both a bike and car are brakes. <br><br>It works great unless you have some physical issue. If you cherished this article therefore you would like to be given more info relating to [http://game.itds.ir/profile/liskurrie Transfering to mountain bike sizing.] generously visit our own website. Find more information about Hardtail mountain bikes here. But older children may have more adventurous tastes, and want for toys that are - well, still miniatures, but are flashier, more expensive, and more functional. People like to ride the mountain bike to those terrains because they enjoy the thrill and the adventure. To find out the latest News and Tips on folding bikes, visit Best Folding Bikes at bestfoldingbikes. <br><br>I ran with Hale and we also tried this method, it was effective. The circumstances will dictate which method to use. Mountain biking amongst the majestic backdrop of the great mountains of Kerala is an unforgettable and thrilling experience. Hence, XC suspension forks tend to have a short travel and are light weight. How do you think people managed decades ago when steel was the only material frames were made of. |
| * Powers of three; no integer having a digit "2" in its ternary representation (2, 5, 6...) can be formed.
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| == Conditions for completeness ==
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| Without loss of generality, assume the sequence ''a''<sub>''n''</sub> is in nondecreasing order, and define the partial sums of ''a''<sub>''n''</sub> as:
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| :<math>s_n=\sum_{m=0}^n a_m</math>.
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| Then the conditions
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| :<math>a_0 = 1 \,</math>
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| :<math>s_{k-1} \ge a_k - 1 \, \forall \, k \ge 1</math>
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| are both necessary and sufficient for ''a''<sub>''n''</sub> to be a complete sequence.<ref name="Honsberger">Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985, pp.123-128.</ref>
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| A corollary to the above states that | |
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| :<math>a_0 = 1\,</math>
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| :<math>2a_k \ge a_{k+1} \, \forall \, k \ge 1</math>
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| are sufficient for ''a''<sub>''n''</sub> to be a complete sequence.<ref name="Honsberger"/> | |
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| However there are complete sequences that do not satisfy this corollary, see {{OEIS|A203074}}.
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| == Other complete sequences ==
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| Below is a list of the better known complete sequences.
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| * The sequence of the number 1 followed by the [[prime number]]s (studied by [[Subbayya Sivasankaranarayana Pillai|S. S. Pillai]]<ref>S. S. Pillai, "An arithmetical function concerning primes", Annamalai University Journal (1930), pp. 159–167.</ref> and others); this follows from [[Bertrand's postulate]].<ref name="Honsberger"/>
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| * The [[Fibonacci numbers]], as well as the Fibonacci numbers with any one number removed.<ref name="Honsberger"/> This follows from the identity that the sum of the first ''n'' Fibonacci numbers is the (''n'' + 2)nd Fibonacci number minus 1 (see [[Fibonacci_numbers#Second_identity]]).
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| * All Fibonacci n-Step numbers,<ref>{{Mathworld|title=Fibonacci n-Step Number|urlname=Fibonaccin-StepNumber}}</ref> where n=2 gives the Fibonacci numbers above, n=3 gives the [[Generalizations of Fibonacci numbers|Tribonacci numbers]] etc. | |
| * The [[Lazy caterer's sequence]] that gives the maximum number of partitions that a plane can be divided into, using n straight lines as dividers.
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| * All higher dimensions of the Lazy caterer's sequence that uses n hyperplanes (of dimension d-1) to maximally divide a space (of dimension d). {{OEIS|id=A216274}}
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| * The Cookie cutter's sequence that gives the maximum number of partitions that a plane can be divided into, using n circles as dividers.<ref>{{Mathworld|title=Plane Division by Circles|urlname=PlaneDivisionbyCircles}}</ref>
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| * All higher dimensions of the Cookie cutter's sequence that uses n hyperspherical surfaces (of dimension d-1) to maximally divide a space (of dimension d). {{OEIS|id=A059250}}
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| == Applications ==
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| Just as the powers of two form a complete sequence due to the binary numeral system, in fact any complete sequence can be used to encode integers as bit strings. The rightmost bit position is assigned to the first, smallest member of the sequence; the next rightmost to the next member; and so on. Bits set to 1 are included in the sum. These representations may not be unique.
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| ===Fibonacci coding===
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| For example, in the '''Fibonacci arithmetic''' system, based on the Fibonacci sequence, the number 17 can be encoded in six different ways:
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| :110111 (F<sub>6</sub> + F<sub>5</sub> + F<sub>3</sub> + F<sub>2</sub> + F<sub>1</sub> = 8 + 5 + 2 + 1 + 1 = 17, maximal form)
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| :111001 (F<sub>6</sub> + F<sub>5</sub> + F<sub>4</sub> + F<sub>1</sub> = 8 + 5 + 3 + 1 = 17) | |
| :111010 (F<sub>6</sub> + F<sub>5</sub> + F<sub>4</sub> + F<sub>2</sub> = 8 + 5 + 3 + 1 = 17)
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| :1000111 (F<sub>7</sub> + F<sub>3</sub> + F<sub>2</sub> + F<sub>1</sub> = 13 + 2 + 1 + 1 = 17)
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| :1001001 (F<sub>7</sub> + F<sub>4</sub> + F<sub>1</sub> = 13 + 3 + 1 = 17)
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| :1001010 (F<sub>7</sub> + F<sub>4</sub> + F<sub>2</sub> = 13 + 3 + 1 = 17, minimal form, as used in [[Fibonacci coding]])
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| :The maximal form above will always use F<sub>1</sub> and will always have a trailing one. The full coding without the trailing one can be found at
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| :{{OEIS|id=A104326}}. Note that by dropping the trailing one, the coding for 17 above occurs as the 16th term of A104326.
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| :The minimal form will never use F<sub>1</sub> and will always have a trailing zero. The full coding without the trailing zero can be found at
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| :{{OEIS|id=A014417}}. This coding is known as the [[Zeckendorf's theorem|Zeckendorf representation]] .
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| In this numeral system, any substring "100" can be replaced by "011" and vice versa due to the definition of the Fibonacci numbers.<ref>{{cite web |url=http://www.goldenmuseum.com/1202FibCdeTransf_engl.html |title=The main operations of the Fibonacci arithmetic |author=Alexey Stakhov |work=Museum of Harmony and Golden Section |accessdate=27 July 2010}}</ref>
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| ===Other coding systems===
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| Other coding systems can be similarly devised. As with the Fibonacci sequence above, these coding systems that employ complete sequences will need to deal with multiple encoding solutions. The two main methods used are the [[greedy algorithm]] that will attempt to minimize the number of terms needed to encode an integer from the complete sequence and a minimizing algorithm that will attempt to maximize the number of terms needed to encode the same integer.
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| * A coding for the sequence of the number 1 followed by the prime numbers using the greedy algorithm can be found at
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| :{{OEIS|id=A007924}}.
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| * A coding for the sequence of the number 1 followed by the prime numbers using a minimizing algorithm can be found at
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| :{{OEIS|id=A205598}}.
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| * A coding for the Lazy caterer's sequence using the greedy algorithm can be found at
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| :{{OEIS|id=A204009}}.
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| ==See also==
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| * [[Ostrowski numeration]]
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| == External links ==
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| * {{MathWorld | urlname=CompleteSequence | title=Complete Sequence}}
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| == References ==
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| <references/>
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| [[Category:Integer sequences]]
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They have powerful and durable mountain bikes with a high-tech suspension system. Macaskill is tackling the Cuillin Ridge in Scotland, not far from where this two-wheeling genius hails from on the Isle of Skye. These bikes are very popular with the sport of mountain biking. Alison Addy is the author of many articles on subjects like mountain bike crashes and published at. With this in mind all you have to do is book the vacation and find the right bike for you and your holiday.
Adjust the tyre pressure too so it is at its optimum level. General Use (Commuting, Mountain Bike, Road Bike, BMX, and Universal Fit) General use helmets are usually about. A wider tire will give you more traction when riding over rocks or in turns, you can run at lower pressure and the more cushioned the ride will feel, but they are heavier. Now please continue on additionally so you can get added info for this topic. A sports drink is better than water as they contain carbohydrates and electrolytes that speed absorption of fluids and can give you an energy boost.
Mountain bikes in Arizona have two great rides in Sedona, Highline and the Dry Creek Loop. * Numbness - Impingement of small nerve branches between the second and third or third and fourth toes can cause swelling that results in numbness, tingling, or burning, or sharp shooting pains into the toes. The whole set-up, including your bag, should weigh no more than 20kgs (44lbs). Once you have answered the above questions and you see all these features present in a bike, you are on your way to choosing the road bike that is right for you. One of the obvious things to check on both a bike and car are brakes.
It works great unless you have some physical issue. If you cherished this article therefore you would like to be given more info relating to Transfering to mountain bike sizing. generously visit our own website. Find more information about Hardtail mountain bikes here. But older children may have more adventurous tastes, and want for toys that are - well, still miniatures, but are flashier, more expensive, and more functional. People like to ride the mountain bike to those terrains because they enjoy the thrill and the adventure. To find out the latest News and Tips on folding bikes, visit Best Folding Bikes at bestfoldingbikes.
I ran with Hale and we also tried this method, it was effective. The circumstances will dictate which method to use. Mountain biking amongst the majestic backdrop of the great mountains of Kerala is an unforgettable and thrilling experience. Hence, XC suspension forks tend to have a short travel and are light weight. How do you think people managed decades ago when steel was the only material frames were made of.