2,4

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 (benoit7848c(AT)orange.fr), Jun 03 2002

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

Also number of different kinds of regular n-gons, one convex, the others self-intersecting. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 20 2005

Contribution from Artur Jasinski (grafix(AT)csl.pl), Oct 28 2008: (Start)

Degree of polynomial which one of the root is Cos[2Pi/n]. These polynomials are:

1: x-1

2: x+1

3: x+1/2

4: x

5: x-1/4

6: -4 + 2 x + x^2

7: x-1/2

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

9: x^2 - 1/2, 1 - 6 x + 8 x^3

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

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

12: x^2 - 3/4

13: -1 + 6 x + 24 x^2 - 32 x^3 - 80 x^4 + 32 x^5 + 64 x^6

14: 1 - 4 x - 4 x^2 + 8 x^3

15: 1 + 8 x - 16 x^2 - 8 x^3 + 16 x^4

16: 1 - 8 x^2 + 8 x^4

17: 1 - 8 x - 40 x^2 + 80 x^3 + 240 x^4 - 192 x^5 - 448 x^6 + 128 x^7 + 256 x^8

18: -1 - 6 x + 8 x^3

19: 1 + 10 x - 40 x^2 - 160 x^3 + 240 x^4 + 672 x^5 - 448 x^6 - 1024 x^7 + 256 x^8 + 512 x^9

5 20: - 20 x^2 + 16 x^4

21: 1 - 16 x + 32 x^2 + 48 x^3 - 96 x^4 - 32 x^5 + 64 x^6

22: -1 + 6 x + 12 x^2 - 32 x^3 - 16 x^4 + 32 x^5

23: -1 - 12 x + 60 x^2 + 280 x^3 - 560 x^4 - 1792 x^5 + 1792 x^6 + 4608 x^7 - 2304 x^8 - 5120 x^9 + 1024 x^10 + 2048 x^11

24: 1 - 16 x^2 + 16 x^4

25: -1 + 10 x + 100 x^2 - 40 x^3 - 800 x^4 + 32 x^5 + 2240 x^6 - 2560 x^8 + 1024 x^10

26: -1 - 6 x + 24 x^2 + 32 x^3 - 80 x^4 - 32 x^5 + 64 x^6

27: 1 + 18 x - 240 x^3 + 864 x^5 - 1152 x^7 + 512 x^9

28: -7 + 56 x^2 - 112 x^4 + 64 x^6

29: -1 + 14 x + 112 x^2 - 448 x^3 - 2016 x^4 + 4032 x^5 + 13440 x^6 - 15360 x^7 - 42240 x^8 + 28160 x^9 + 67584 x^10 - 24576 x^11 - 53248 x^12 + 8192 x^13 + 16384 x^14

30: 1 - 8 x - 16 x^2 + 8 x^3 + 16 x^4

etc. All polynomials which one of the roots is Cos[2Pi/n] (for rational n)

belonging to solvable Galois groups, what mean that are available to express by radicals. (End)

G. Polya and G. Szego, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part Eight, Chap. 1, Sect. 6, Problems 60&61.

T. D. Noe, Table of n, a(n) for n=2..10000

K. S. Brown, The Half-Totient Tree

Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.

Eric Weisstein's World of Mathematics, Polygon Triangle Picking

Eric Weisstein's World of Mathematics, Trigonometry Angles

phi(n)/2 for n >= 3.

a(n) = Sum(k/n: 1<=k<n and GCD(n, k)=1) = A023896(n)/n for n>2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 20 2005

Table[ EulerPhi[n]/2, {n, 3, 50}]

Cf. A000010, A055684, A046657, A049806, A049703, A062956.

Adjacent sequences: A023019 A023020 A023021 this_sequence A023023 A023024 A023025

Sequence in context: A070804 A104481 A078709 this_sequence A100677 A083290 A121842

nonn

David W. Wilson (davidwwilson(AT)comcast.net)