cp's OEIS Frontend

A subsequence of A000961 without numbers divisible by 4.

The powers of odd primes are given in A061345 (with offset 0).

These Markoff numbers (see A002559) have been proved to obey the Frobenius-Markoff uniqueness conjecture. See Aigner, Corollary 3.20, p. 59, and there the references [4] A. Baragar, [18] J. O. Button, and [119] Ying Zhang.

The number of (x, y) pairs in row n is 1 for n = 1 and 2, and 2^P, with P = P1 + P7, where P1 and P7 are the number of prime factors 1 modulo 8 and 7 modulo 8, respectively, of A057126(n), for n >= 3.

See A057126 for comments concerning its representation by x^2 - 2*y^2.

The numbers A057126 are given by 2^e_2 * Product_{i=1..P1} p_{1,i}^e_{1,i} * Product_{j=1..P7} p_{7,j}^e_{7,j}, with the odd primes p_{1,i} and p_{7,j} congruent to 1 and 7 modulo 8, respectively. See A007519 and A007522 for these odd primes. Together with 2 these primes are given in A038873, and without 2 in A001132. The exponents are e_2 = 0 or 1, and e_{1,i} and e_{7,j} are nonnegative. The a(1) = 1 is obtained if all exponents vanish. For the proof see Lemma 18 of the linked W. Lang paper, pp. 22 - 23.

The general solutions are obtained from each fundamental solution by application of integer powers of the matrix Auto' = Mat([3,4], [2,3]). See the linked paper eq (28), p. 14, and eq. (40), p. 17 for D = 2, and k = A057126(n). For the explicit form of the powers of Auto' in terms of Chebyshev polynomials S(n, 6) = A001109(n+1) see there eq. (38), and Lemma 10, eq. (43), p. 17.

The conversion to the pair of proper solutions (X, Y) of X^2 - 2*Y^2 = A057126(n) is given by (X, Y) = (2*y - x, x - y). This may result in solutions with negative Y values. They are then transformed to the fundamental positive proper solutions via the mentioned matrix Auto'. See the right part of the example below. For this conversion see also the Nov 09 2009 comment in A035251 by Franklin T. Adams-Watters.

This constant c gives sqrt(11)*rho_1, where rho_1 = length(Z, D_1), with the fixed point Z = -(A385445 + A385446*i)/(2*11) of the complex map w given in A385445 and D_1 = i.

See A385445 for details and the linked paper, eq. (7b).

This constant c gives the real part of -2*11*Z = (c + d*i), where Z is the (finite) fixed point of a complex function w (of the loxodromic type) mapping iteratively the vertices of golden triangles, starting with vertices (D_1, D_2, D_3), circumscribed by the unit circle with center at the origin, and D_1 = i, D_2 = (s - phi*i)/2 and D_3 = (-s - phi*i)/2. This function is w(z) = a*z + b, with a = (-1 + phi) * exp(-(3*Pi/5)*i) = -((2 - phi) + s*i)/2 and b = (1 - phi)*i, where s = sqrt(3 - phi) = A182007 (the length of the base (D_2, D_3) of the first triangle, also called s_1).

The imaginary part of -2*11*Z is d = -7 + 10*phi = A385446.

If the fixed point Z = -(0.20594... + 0.41728...*i) is chosen as origin then the loxodromic map is W(z') = a*z' (where z' = z - Z and W(z') = w(z'+Z) - Z).

For details see the linked paper, eqs.(5a,b) for w(z), eq.(6) for Z and eq.(7) for W(z'). (In eq.(5b) the i is missing in the exponent.) The nesting of golden triangles as shown in Fig. 1 of the link leads to the fixed point Z.

The vertices of the nested golden triangles can be connected by a spiral built of circular sections with angle 108 degrees, centered at vertices D_{n+3} and shrinking radii s_n =(- 1 + phi)^(n-1)*s. Note that the curvature of this spiral is not continuous.

The length(Z, D_n) =: rho_n of the spokes of the spiral is (-1 + phi)^(n-1)*rho_1, with sqrt(11)*rho_1 = sqrt(8 + 9*phi) = sqrt(5 + 7*phi)*s = A385447.

For the length ratio rho_1/s see A385448.

The logarithmic spiral connecting the vertices D_{n+1} is given in polar coordinates by rho(Phi) = rho_1 * exp((-(5/(3*Pi)) * log(phi)*Phi), with the vertices obtained in polar coordinates for Phi = Phi_n = (3*Pi/5)*n, namely rho(Phi_n) = rho_{n+1}, for n >= 0. For log(phi) see A002390. Note that the nonnegative x-axis is now along Z, D_1. The angle(Z, D1, D4) =: gamma is given by arctan((18 - 11*phi)/s) = arcsin(rho_4 / 2) = 0.169860704... or 9.7323... degrees. See Fig. 3 of the linked paper.

This constant d gives the imaginary part of -2*11*Z = c + d*i, where Z is the fixed point of a complex function w (of the loxodromic type) mapping vertices of golden triangles, starting with vertices (D_1, D_2, D_3), circumcribed by the unit circle with center at the origin, and D_1 = i (the complex unit), D_2 = (s - phi*i)/2 and D_3 = (-s - phi*i)/2. This function is w(z) = a*z + b, with a = (-1 + phi) * exp(-3*Pi*i/5) = -((2 - phi) + s*i)/2 and b = (1 - phi)*i, where s = sqrt(3 - phi) = A182007 (the length of the base (D2, D3) of the first triangle).

The real part c = (-1 + 3*phi)*s is given in A385445.

For details see A385445, and eqs.(5a,b) of the linked paper there.

This equals the ratio length(Z, D_1)/s, with the fixed point of a complex loxodromic map w mapping iteratively golden triangles, starting with the one inscribed in a circumcircle with center ot the origin of the complex plane, the top vertex D_1 = i (the complex unit) and the base D_2 = (s - phi*i)/2, D_3 = (-s - phi*i)/2, with s = A182007.

See A385445 for details and a linked paper.

This sequence gives the number of the so-called king permutations, for n >= 1, counted in A002464, that satisfy the additional restriction |p_n - p_1| = 0 or 1.

Only two rows, each of length 10.

This array gives the deciduous (baby, milk) tooth numbering systems for humans, according to the FDI two-digit notation, and IS0 3950.

For the permanent teeth numbering for humans see A384919.

Only two rows, each of length 16.

This array gives the permanent tooth numbering systems for humans, according to the FDI two-digit notation, and IS0 3950.

For the deciduous (baby, milk) tooth numbering system for humans see A384920.

The number of such permutations of [n] is 1 for n = 1 (the p(i) condition is not needed), and 0 for n = 2, 3 and 4, hence a(1) = 1/4 and a(n) = 0 for n = 2, 3 and 4.

The number of permutations of [n] with |p(i+1) - p(i)| >= 2, for i = 1..(n-1), for n >= n is given by A002464(n), for n >= 0. See also A001266(n) = A002464(n)/2, for n >= 2. These permutations are also called king permutations, e.g., in A382644.