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Centered Pentagonal Number
A centered pentagonal number is a centered figurate number that represents a pentagon with a dot in the center and all other dots surrounding the center in successive pentagonal layers. The centered pentagonal number for ''n'' is given by the formula :P_=, n\geq1 The first few centered pentagonal numbers are 1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526, 601, 681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031, 2176, 2326, 2481, 2641, 2806, 2976 . Properties *The parity of centered pentagonal numbers follows the pattern odd-even-even-odd, and in base 10 the units follow the pattern 1-6-6-1. *Centered pentagonal numbers follow the following Recurrence relations: :P_=P_+5n , P_0=1 :P_=3(P_-P_)+P_ , P_0=1,P_1=6,P_2=16 *Centered pentagonal numbers can be expressed using Triangular Numbers: :P_=5T_+1 See also *Pentagonal number *Polygonal number *Centered polygonal number The centered polygonal numbers are a class of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
226 (number)
226 (two hundred ndtwenty-six) is the natural number following 225 and preceding 227. In mathematics 226 is a happy number, and a semiprime ( 2× 113), and a member of Aronson's sequence. At most 226 different permutation pattern In combinatorial mathematics and theoretical computer science, a permutation pattern is a sub-permutation of a longer permutation. Any permutation may be written in one-line notation as a sequence of digits representing the result of applying the p ...s can occur within a single 9-element permutation. In other fields * The number of ages Hanako has been alive. References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygonal Number
In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers. Definition and examples The number 10 for example, can be arranged as a triangle (see triangular number): : But 10 cannot be arranged as a square (geometry), square. The number 9, on the other hand, can be (see square number): : Some numbers, like 36, can be arranged both as a square and as a triangle (see square triangular number): : By convention, 1 is the first polygonal number for any number of sides. The rule for enlarging the polygon to the next size is to extend two adjacent arms by one point and to then add the required extra sides between those points. In the following diagrams, each extra layer is shown as in red. Triangular numbers : Square numbers : Polygons with higher numbers of sides, such as pentagons and hexagons, can also be constructe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The ''n''th pentagonal number ''pn'' is the number of ''distinct'' dots in a pattern of dots consisting of the ''outlines'' of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. ''p''n is given by the formula: :p_n = =\binom+3\binom for ''n'' ≥ 1. The first few pentagonal numbers are: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the natural numbers from 1 to . The sequence of triangular numbers, starting with the 0th triangular number, is (This sequence is included in the On-Line Encyclopedia of Integer Sequences .) Formula The triangular numbers are given by the following explicit formulas: T_n= \sum_^n k = 1+2+3+ \dotsb +n = \frac = , where \textstyle is a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The first equation can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-square" arrangement of objects corresponding to the triangular numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Relations
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
601 (number)
600 (six hundred) is the natural number following 599 and preceding 601. Mathematical properties Six hundred is a composite number, an abundant number, a pronic number and a Harshad number. Credit and cars * In the United States, a credit score of 600 or below is considered poor, limiting available credit at a normal interest rate. * NASCAR runs 600 advertised miles in the Coca-Cola 600, its longest race. * The Fiat 600 is a car, the SEAT 600 its Spanish version. Integers from 601 to 699 600s * 601 = prime number, centered pentagonal number * 602 = 2 × 7 × 43, nontotient, number of cubes of edge length 1 required to make a hollow cube of edge length 11, area code for Phoenix, AZ along with 480 and 623 * 603 = 32 × 67, Harshad number, Riordan number, area code for New Hampshire * 604 = 22 × 151, nontotient, totient sum for first 44 integers, area code for southwestern British Columbia (Lower Mainland, Fraser Valley, Sunshine Coast and Sea to Sky) * 605 = 5 × 112, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
526 (number)
500 (five hundred) is the natural number following 499 and preceding 501. Mathematical properties 500 = 22 × 53. It is an Achilles number and an Harshad number, meaning it is divisible by the sum of its digits. It is the number of planar partitions of 10. Other fields Five hundred is also *the number that many NASCAR races often use at the end of their race names (e.g., Daytona 500), to denote the length of the race (in miles, kilometers or laps). *the longest advertised distance (in miles) of the IndyCar Series and its premier race, the Indianapolis 500. Slang names * Monkey (UK slang for £500; USA slang for $500) Integers from 501 to 599 500s 501 501 = 3 × 167. It is: * the sum of the first 18 primes (a term of the sequence ). * palindromic in bases 9 (6169) and 20 (15120). 502 * 502 = 2 × 251 * vertically symmetric number 503 503 is: * a prime number. * a safe prime. * the sum of three consecutive primes (163 + 167 + 173). * the sum of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
456 (number)
400 (four hundred) is the natural number following 399 and preceding 401. Mathematical properties 400 is the square of 20. 400 is the sum of the powers of 7 from 0 to 3, thus making it a repdigit in base 7 (1111). A circle is divided into 400 grads, which is equal to 360 degrees and 2π radians. (Degrees and radians are the SI accepted units). 400 is a self number in base 10, since there is no integer that added to the sum of its own digits results in 400. On the other hand, 400 is divisible by the sum of its own base 10 digits, making it a Harshad number. Other fields Four hundred is also * The Four Hundred (oligarchy) of ancient Athens. * An HTTP status code for a bad client request. * The Four Hundred (sometimes The Four Hundred Club) a phrase meaning the wealthiest, most famous, or most powerful social group (see, e.g., Ward McAllister), leading to the generation of such lists as the Forbes 400. * The Atari 400 home computer. * A former limited stop bus route whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
391 (number)
300 (three hundred) is the natural number following 299 and preceding 301. Mathematical properties The number 300 is a triangular number and the sum of a pair of twin primes (149 + 151), as well as the sum of ten consecutive primes (13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47). It is palindromic in 3 consecutive bases: 30010 = 6067 = 4548 = 3639, and also in base 13. Factorization is 30064 + 1 is prime Other fields Three hundred is: * In bowling, a perfect score, achieved by rolling strikes in all ten frames (a total of twelve strikes) * The lowest possible Fair Isaac credit score * Three hundred ft/s is the maximum legal speed of a shot paintball * In the Hebrew Bible, the size of the military force deployed by the Israelite judge Gideon against the Midianites () * According to Islamic tradition, 300 is the number of ancient Israeli king Thalut's soldiers victorious against Goliath's soldiers * According to Herodotus, 300 is the number of ancient Spart ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
331 (number)
300 (three hundred) is the natural number following 299 and preceding 301. Mathematical properties The number 300 is a triangular number and the sum of a pair of twin primes (149 + 151), as well as the sum of ten consecutive primes (13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47). It is palindromic in 3 consecutive bases: 30010 = 6067 = 4548 = 3639, and also in base 13. Factorization is 30064 + 1 is prime Other fields Three hundred is: * In bowling, a perfect score, achieved by rolling strikes in all ten frames (a total of twelve strikes) * The lowest possible Fair Isaac credit score * Three hundred ft/s is the maximum legal speed of a shot paintball * In the Hebrew Bible, the size of the military force deployed by the Israelite judge Gideon against the Midianites () * According to Islamic tradition, 300 is the number of ancient Israeli king Thalut's soldiers victorious against Goliath's soldiers * According to Herodotus, 300 is the number of ancient Spar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
276 (number)
276 (two hundred ndseventy-six) is the natural number following 275 and preceding 277. In mathematics 276 is the sum of 3 consecutive fifth powers (276 = 15 + 25 + 35). As a figurate number it is a triangular number, a hexagonal number, and a centered pentagonal number, the third number after 1 and 6 to have this combination of properties. 276 is the size of the largest set of equiangular lines in 23 dimensions. The maximal set of such lines, derived from the Leech lattice, provides the highest dimension in which the "Gerzon bound" of \binom is known to be attained; its symmetry group is the third Conway group, Co3. 276 is the smallest number for which it is not known if the corresponding aliquot sequence either terminates or ends in a repeating cycle. In other fields In the Christian calendar, there are 276 days from the Annunciation on March 25 to Christmas on December 25, a number considered significant by some authors. See also *The years 276 and 276 BC __NOTOC__ Yea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
