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Consider an (0, 1)-matrix such as(1)for . Call two elements adjacent if they lie in positions and , and , or and for some . Call the number of such arrays with no pairs of adjacent 1s. Equivalently, is the number of configurations of nonattacking kings on an chessboard with regular hexagonal cells.The first few values of for , 2, ... are 2, 6, 43, 557, 14432, ... (OEIS A066863).The hard square hexagon constant is then given by(2)(3)(OEIS A085851).Amazingly, is algebraic and is given by(4)where(5)(6)(7)(8)(9)(10)(11)(Baxter 1980, Joyce 1988ab).The variable can be expressed in terms of the tribonacci constant(12)where is a polynomial root, as(13)(14)(15)(T. Piezas III, pers. comm., Feb. 11, 2006).Explicitly, is the unique positive root(16)where denotes the th root of the polynomial in the ordering of the Wolfram Language...

The tribonacci numbers are a generalization of the Fibonacci numbers defined by , , , and the recurrence equation(1)for (e.g., Develin 2000). They represent the case of the Fibonacci n-step numbers.The first few terms using the above indexing convention for , 1, 2, ... are 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, ... (OEIS A000073; which however adopts the alternate indexing convention and ).The first few prime tribonacci numbers are 2, 7, 13, 149, 19341322569415713958901, ... (OEIS A092836), which have indices 3, 5, 6, 10, 86, 97, 214, 801, 4201, 18698, 96878, ... (OEIS A092835), and no others with (E. W. Weisstein, Mar. 21, 2009).Using Brown's criterion, it can be shown that the tribonacci numbers are complete; that is, every positive number can be written as the sum of distinct tribonacci numbers. Moreover, every positive number has a unique Zeckendorf-like expansion as the sum of distinct tribonacci numbers and that sum does..

The tetranacci numbers are a generalization of the Fibonacci numbers defined by , , , , and the recurrence relation(1)for . They represent the case of the Fibonacci n-step numbers. The first few terms for , 1, ... are 0, 1, 1, 2, 4, 8, 15, 29, 56, 108, 208, ... (OEIS A000078).The first few prime tetranacci numbers have indices 3, 7, 11, 12, 36, 56, 401, 2707, 8417, 14096, 31561, 50696, 53192, 155182, ... (OEIS A104534), corresponding to 2, 29, 401, 773, 5350220959, ... (OEIS A104535), with no others for (E. W. Weisstein, Mar. 21, 2009).An exact expression for the th tetranacci number for can be given explicitly by(2)where the three additional terms are obtained by cyclically permuting , which are the four roots of the polynomial(3)Alternately,(4)This can be written in slightly more concise form as(5)where is the th root of the polynomial(6)and and are in the ordering of the Wolfram Language's Root object.The tetranacci numbers..

The hexanacci numbers are a generalization of the Fibonacci numbers defined by , , , , , , and the recurrence relation(1)for . They represent the case of the Fibonacci n-step numbers.The first few terms for , 2, ... are 1, 1, 2, 4, 8, 16, 32, 63, 125, 248, ... (OEIS A001592).An exact formula for the th hexanacci number can be given explicitly in terms of the six roots of(2)as(3)The ratio of adjacent terms tends to the positive root of , namely 1.98358284342... (OEIS A118427), sometimes called the hexanacci constant.

The snub cube, also called the cubus simus (Kepler 1619, Weissbach and Martini 2002) or snub cuboctahedron, is an Archimedean solid having 38 faces (32 triangular and 6 square), 60 edges, and 24 vertices. It is a chiral solid, and hence has two enantiomorphous forms known as laevo (left-handed) and dextro (right-handed).It is Archimedean solid , uniform polyhedron , and Wenninger model . It has Schläfli symbol and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["SnubCube"].Surprisingly, the tribonacci constant is intimately related to the metric properties of the snub cube.It can be constructed by snubification of a unit cube with outward offset(1)(2)and twist angle(3)(4)(5)(6)Here, the notation indicates a polynomial root and is the tribonacci constant.An attractive dual of the two enantiomers superposed on one another is illustrated above.Its dual polyhedron is the pentagonalicositetrahedron.The..

The pentagonal icositetrahedron is the 24-faced dual polyhedron of the snub cube and Wenninger dual . The mineral cuprite () forms in pentagonal icositetrahedral crystals (Steinhaus 1999, pp. 207 and 209).Because it is the dual of the chiral snub cube, the pentagonal icositetrahedron also comes in two enantiomorphous forms, known as laevo (left) and dextro (right). An attractive dual of the two enantiomers superposed on one another is illustrated above.A cube, octahedron, and stella octangula can all be inscribed on the vertices of the pentagonal icositetrahedron (E. Weisstein, Dec. 25, 2009).Surprisingly, the tribonacci constant is intimately related to the metric properties of the pentagonal icositetrahedron cube.Its irregular pentagonal faces have vertex angles of(1)(2)(3)(four times) and(4)(5)(6)(once), where is a polynomial root and is the tribonacci constant.The dual formed from a snub cube with..

The heptanacci numbers are a generalization of the Fibonacci numbers defined by , , , , , , , and the recurrence relation(1)for . They represent the case of the Fibonacci n-step numbers.The first few terms for , 2, ... are 1, 1, 2, 4, 8, 16, 32, 64, 127, 253, ... (OEIS A066178).An exact formula for the th heptanacci number can be given explicitly in terms of the seven roots of(2)as(3)The ratio of adjacent terms tends to the real root of , namely 1.99196419660... (OEIS A118428), sometimes called the heptanacci constant.

The pentanacci numbers are a generalization of the Fibonacci numbers defined by , , , , , and the recurrence relation(1)for . They represent the case of the Fibonacci n-step numbers.The first few terms for , 2, ... are 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, ... (OEIS A001591).The ratio of adjacent terms tends to the real root of , namely 1.965948236645485... (OEIS A103814), sometimes called the pentanacci constant.An exact formula for the th pentanacci number can be given explicitly in terms of the five roots of(2)as(3)The pentanacci numbers have generating function(4)

The silver constant is the algebraic number givenby(1)(2)(3)(OEIS A116425), where denotes a polynomial root.Defining the nested radical expression(4)the silver constant is given by(5)(T. Piezas, pers. comm., Feb. 16, 2006).The silver constant is the seventh Beraha constant. Surprisingly, it also appears in the 3-cycle of the logistic map.

The plastic constant , sometimes also called the silver number or plastic number, is the limiting ratio of the successive terms of the Padovan sequence and Perrin sequence. It is given by(1)(2)(3)(OEIS A060006), where denotes a polynomial root. It is therefore an algebraic number of degree 3.It is also given by(4)where(5)where is the -function and the half-period ratio is equal to .The plastic constant was originally studied in 1924 by Gérard Cordonnier when he was 17. In his later correspondence with Dom Hans van der Laan, he described applications to architecture, using the name "radiant number." In 1958, Cordonnier gave a lecture tour that illustrated the use of the constant in many existing buildings and monuments (C. Mannu, pers comm., Mar. 11, 2006). satisfies the algebraic identities(6)and(7)and is therefore is one of the numbers for which there exist natural numbers and such that and . It was proven..

As a consequence of Matiyasevich's refutation of Hilbert's 10th problem, it can be proved that there does not exist a general algorithm for solving a general quartic Diophantine equation. However, the algorithm for constructing such an unsolvable quartic Diophantine equation can require arbitrarily many variables (Matiyasevich 1993).As a part of the study of Waring's problem, it is known that every positive integer is a sum of no more than 19 positive biquadrates (), that every "sufficiently large" integer is a sum of no more than 16 positive biquadrates (), and that every integer is a sum of at most 10 signed biquadrates (; although it is not known if 10 can be reduced to 9). The first few numbers which are a sum of four fourth powers ( equations) are 353, 651, 2487, 2501, 2829, ... (OEIS A003294).The 4.1.2 equation(1)is a case of Fermat's last theorem with and therefore has no solutions. In fact, the equations(2)also have no solutions..

As a part of the study of Waring's problem, it is known that every positive integer is a sum of no more than 9 positive cubes (), that every "sufficiently large" integer is a sum of no more than 7 positive cubes (; although it is not known if 7 can be reduced), and that every integer is a sum of at most 5 signed cubes (; although it is not known if 5 can be reduced to 4).It is known that every can be written in the form(1)An elliptic curve of the form for an integer is known as a Mordell curve.The 3.1.2 equation(2)is a case of Fermat's last theorem with . In fact, this particular case was known not to have any solutions long before the general validity of Fermat's last theorem was established. Thue showed that a Diophantine equation of the form(3)for , , and integers, has only finite many solutions (Hardy 1999, pp. 78-79).Miller and Woollett (1955) and Gardiner et al. (1964) investigated integersolutions of(4)i.e., numbers representable as the..

A general quadratic Diophantine equation in two variables and is given by(1)where , , and are specified (positive or negative) integers and and are unknown integers satisfying the equation whose values are sought. The slightly more general second-order equation(2)is one of the principal topics in Gauss's Disquisitiones arithmeticae. According to Itô (1987), equation (2) can be solved completely using solutions to the Pell equation. In particular, all solutions of(3)are among the convergents of the continued fractions of the roots of .Solution to the general bivariate quadratic Diophantine equation is implemented in the Wolfram Language as Reduce[eqn && Element[x|y, Integers], x, y].For quadratic Diophantine equations in more than two variables, there exist additional deep results due to C. L. Siegel.An equation of the form(4)where is an integer is a very special type of equation called a Pell equation...

A -multigrade equation is a Diophantine equation of the form(1)for , ..., , where and are -vectors. Multigrade identities remain valid if a constant is added to each element of and (Madachy 1979), so multigrades can always be put in a form where the minimum component of one of the vectors is 1.Moessner and Gloden (1944) give a bevy of multigrade equations. Small-order examples are the (2, 3)-multigrade with and :(2)(3)the (3, 4)-multigrade with and :(4)(5)(6)and the (4, 6)-multigrade with and :(7)(8)(9)(10)(Madachy 1979).A spectacular example with and is given by and (Guy 1994), which has sums(11)(12)(13)(14)(15)(16)(17)(18)(19)Rivera considers multigrade equations involving primes, consecutive primes, etc.Analogous multigrade identities to Ramanujan's fourth power identity of form(20)can also be given for third and fifth powers, the former being(21)with , 2, 3, for any positive integer , and where(22)(23)and the one for fifth..

Using a Tschirnhausen transformation, the principal quintic form can be transformed to the one-parameter form(1)named after Francesco Brioschi (1824-1897) and which is important to the Klein's solution of the general quintic in terms of hypergeometric functions (Doyle and McMullen). This can be attained by using the transformation,(2)(Dickson 1959) and eliminating the variable between the two using resultants to form a new quintic(3)where(4)(5)(6)Equating coefficients with a generic principal quintic(7)results in a system of three equations in the three unknowns , , and . Amazingly, this can be resolved to a single equation that is only a quadratic and given in the variable by(8)(Dickson 1959).

Expressions of the form(1)are called nested radicals. Herschfeld (1935) proved that a nested radical of real nonnegative terms converges iff is bounded. He also extended this result to arbitrary powers (which include continued square roots and continued fractions as well), a result is known as Herschfeld's convergence theorem.Nested radicals appear in the computation of pi,(2)(Vieta 1593; Wells 1986, p. 50; Beckmann 1989, p. 95), in trigonometrical values of cosine and sine for arguments of the form , e.g.,(3)(4)(5)(6)Nest radicals also appear in the computation of the goldenratio(7)and plastic constant(8)Both of these are special cases of(9)which can be exponentiated to give(10)so solutions are(11)In particular, for , this gives(12)The silver constant is related to the nested radicalexpression(13)There are a number of general formula for nested radicals (Wong and McGuffin). For example,(14)which gives as special..

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