cp's OEIS Frontend

The number of distinct linear fractional transformations of order n. Also the half-totient function can be used to construct a tree containing all the integers. On the zeroth rank we have just the integers 1 and 2: immediate "ancestors" of 1 and 2 are (1: 3,4,6 2: 5,8,10,12) etc. - Benoit Cloitre, Jun 03 2002

Moebius transform of floor(n/2). - Paul Barry, Mar 20 2005

Also number of different kinds of regular n-gons, one convex, the others self-intersecting. - Reinhard Zumkeller, Aug 20 2005

From Artur Jasinski, Oct 28 2008: (Start)

Degrees of minimal polynomials of cos(2*Pi/n). The first few are

1: x - 1

2: x + 1

3: 2*x + 1

4: x

5: 4*x^2 + 2*x - 1

6: 2*x - 1

7: 8*x^3 + 4*x^2 - 4*x - 1

8: 2*x^2 - 1

9: 8*x^3 - 6*x + 1

10: 4*x^2 - 2*x - 1

11: 32*x^5 + 16*x^4 - 32*x^3 - 12*x^2 + 6*x + 1

These polynomials have solvable Galois groups, so their roots can be expressed by radicals. (End)

a(n) is the number of rationals p/q in the interval [0,1] such that p + q = n. - Geoffrey Critzer, Oct 10 2011

It appears that, for n > 2, a(n) = A023896(n)/n. Also, it appears that a record occurs at n > 2 in this sequence if and only if n is a prime. For example, records occur at n=5, 7, 11, 13, 17, ..., all of which are prime. - John W. Layman, Mar 26 2012

From Wolfdieter Lang, Dec 19 2013: (Start)

a(n) is the degree of the algebraic number of s(n)^2 = (2*sin(Pi/n))^2, starting at a(1)=1. s(n) = 2*sin(Pi/n) is the length ratio side/R for a regular n-gon inscribed in a circle of radius R (in some length units). For the coefficient table of the minimal polynomials of s(n)^2 see A232633.

Because for even n, s(n)^2 lives in the algebraic number field Q(rho(n/2)), with rho(k) = 2*cos(Pi/k), the degree is a(2*l) = A055034(l). For odd n, s(n)^2 is an integer in Q(rho(n)), and the degree is a(2*l+1) = A055034(2*l+1) = phi(2*l+1)/2, l >= 1, with Euler's totient phi=A000010 and a(1)=1. See also A232631-A232633.

(End)

Also for n > 2: number of fractions A182972(k)/A182973(k) such that A182972(k) + A182973(k) = n, A182972(n) and A182973(n) provide an enumeration of positive rationals < 1 arranged by increasing sum of numerator and denominator then by increasing numerator. - Reinhard Zumkeller, Jul 30 2014

Number of distinct rectangles with relatively prime length and width such that L + W = n, W <= L. For a(17)=8; the rectangles are 1 X 16, 2 X 15, 3 X 14, 4 X 13, 5 X 12, 6 X 11, 7 X 10, 8 X 9. - Wesley Ivan Hurt, Nov 12 2017

After including a(1) = 1, the number of elements of any reduced residue system mod* n used by Brändli and Beyne is a(n). See the examples below. - Wolfdieter Lang, Apr 22 2020

a(n) is the number of ABC triples with n = c. - Felix Huber, Oct 12 2023

The length of row l of this table is delta(l) + 1 = A055034(l) + 1, l >= 1, that is: 2, 2, 2, 3, 3, 3, 4, 5, 4, 5, 6, 5, 7, 7, 5, ...

s(n):= 2*sin(Pi/n) is the length ratio side/R of a regular n-gon inscribed in a circle of radius R (in some length units). In general s(n)^2 = 4 - rho(n)^2 with rho(n):= 2*cos(Pi/n), the length ratio (smallest diagonal)/s(n) in the regular n-gon (n>=2). If n is even, say 2*l, l>=1, then s(2*l)^2 = 2 - rho(l) (because rho(2*l)^2 = rho(l) + 2). Therefore s(2*l) is an integer in the algebraic number field Q(rho(l)).

Its (monic) minimal polynomial is obtained from the conjugates of rho(l), called rho(l;j), j = 1, 2, ..., delta(l), which are the zeros of the minimal polynomial of rho(l) = rho(l;1) of degree delta(l) = A055034(l), called C(l, x) in A187360. These conjugates are therefore rho(l;j) = 2*cos(Pi*rpnodd(l,j)/l) where rpnodd(l,j) is the j-th entry of the list rpnodd(l) of the odd numbers < l which are relatively prime to l (for example, rpnodd(9) = [1,5,7], and rpnodd(9,2) = 5). From this the conjugates of s(2*l)^2 become 2 - rho(l;j), and the minimal polynomial of s(2*l)^2 is MPs2(l, x) = product( x - (2- rho(l;j)), j=1..delta(l)) for l >=1. Because the zeros of C(l, x) are integers in the algebraic number field Q(rho(l)) written in its power basis (see table 4 of the link under A187360 to the Q(2 cos(Pi/n)) paper) one finds, after expansion and reducing powers of rho(l) modulo C(l, rho(l)), directly the integer coefficients appropriate for this (monic) minimal polynomial. Only the equation C(l, rho(l)) = 0 is needed, not the trigonometric version of rho(l) and its powers.

This computation was motivated by a preprint of S. Mustonen, P. Haukkanen and J. K. Merikoski, called 'Polynomials associated with squared diagonals of regular polygons', Nov 16 2013.

The length of row l is delta(2*l+1) + 1 = A055034(2*l+1) + 1, l >= 0.

See the comments on A232631 (even n case) for s(n) = 2*sin(Pi/n) and the minimal polynomial of s(n)^2. Here n = 2*l+1 and s(2*l+1)^2 = 4 - rho(2*l+1)^2 is an integer in the algebraic number field Q(rho(2*l+1)). The minimal polynomial of s(2*l+1)^2 is then MPs2(2*l+1, x) = product(x - 2*(1 + cos(Pi*rpnodd(2*l+1,j)/(2*l+1))), j=1..delta(2*l+1)), l >= 0, where rpnodd(2*l+1) is the list of positive odd numbers < 2*l+1 and relatively prime to 2*l+1. rpnodd(2*l+1,j) is the j-th member of this increasingly ordered list. Here the identity 4 - (2*cos(Pi*(2*k+1)/(2*l+1)))^2 = 2*(1 - cos(Pi*2*(2*k+1)/(2*l+1))) = 2*(1 - (- cos(Pi*(2*l+1 - 2*(2*k+1))/ (2*l+1)))) has been used, and for 2*k+1 < 2*l+1 and gcd(2*k+1, 2*l+1) = 1 this becomes the product given above because 1 = gcd(-(2*k+1), 2*l+1) = gcd(-2*(2*k+1), 2*l+1) = gcd(2*l+1, -2*(2*k+1) + (2*l+1)).

This computation was motivated by a preprint of S. Mustonen, P. Haukkanen and J. K. Merikoski, called ``Polynomials associated with squared diagonals of regular polygons'', Nov 16 2013.

The length of row n is A023022(n) + 1, with A023022(1) = 1.

For the minimal polynomials of 2*sin(Pi/n) see A232633 (n >= 1), A232632 (even n) and A232631 (odd n).

The degree of the algebraic number over the rationals rho(n) = 2*cos(Pi/n) is delta(n) = A055034(n). The degree of rho(n)^2, for n = 2*m, is delta(m), for m >= 1. This is due to the trigonometric identity (half-angle formula) rho(2*m)^2 = 2 + rho(m). For m >= 0 rho(2*m+1)^2 has degree delta(2*l+1).

For the field extension Q(rho(n)) see the W. Lang link where the minimal polynomial of rho(n), named C(n, x), is shown in Table 2. See also A187360.

In both cases the conjugates (over Q) of rho(n), that is the roots of the minimal polynomial C(n, x) enter. This is the set with elements 2*cos(Pi*(2*m+1)/n) = R(2*m+1, rho(n)), for m from {0..floor((n-1)/2)} with gcd(2*m + 1, n) = 1. The polynomial R(n, x) = 2*T(n, x/2) is a monic version of the Chebyshev T polynomials; see A127672 for its coefficients. This list of numbers 2*m+1 is named rpnodd(n) (e.g., n = 12, rpnodd(n) = [1, 5, 7, 11]). #rpnodd(n) = delta(n). The conjugates of rho(n) are then rho(n; j) = 2*cos(Pi*rpnodd(n)_j/n), for j = 1, 2, ..., delta(n), and rho(n; 1) = rho(n), for n >= 2. Because rpnodd(1) is the empty set, a separate case is needed, namely rho(1; 1) = -2.

The minimal polynomials for rho(2*m)^2 are then MPc2(m, x) = Product_{j=1..delta(m)} (x - (2 + rho(m; j)) = Product_{j=1..delta(m)} (x - (2 + R(rpnodd(m)_j, rho(m)))) for m >= 2, and MPc2(1, x) = x. But because C(m, rho(m)) = 0, this has to be evaluated modulo this minimal polynomial of rho(m), that is all powers rho(m)^k with k >= delta(m) are replaced, leaving elements of Q(rho(m)) written in its power basis. Note that the trigonometric form of rho(m) is not used in this computation.

In the odd n case one uses for the conjugates of rho(2*m+1)^2 the formula R(2*m+1, x)^2 = R(2*(2*m+1), x) + 2, obtained from the product formula for R(n, x)*R(k, x) = R(n+m, x) + 2. Then for the reduced 2*m+1 values defined above R(2*(2*m+1), x) + 2 can be replaced by -R(rpnodd(2*m+1)j, x) + 2, for j = 1, ..., delta(2*m+1). Thus MPc2(2*m+1, x) = Product{j=1..delta(2*m+1)} (x - (2 - R(rpnodd(2*m+1)_j, x)), for m >= 1. But for m = 0 (n = 1) the degree of rho(1)^2 = (-2)^2 is 1, hence MPc2(1, x) = x - 4.

These polynomials appear, e.g., in the Salas and Sokal paper, see Table 1, p. 64, or p. 620, for n = 2..16, where rho(n)^2 are called Beraha numbers B_n. I was informed about this paper by Gary W. Adamson.