OFFSET
1,2
COMMENTS
The n-th term is the number of notes of the (2n-1)-limit tonality diamond. This is a term from music theory and means the scale consisting of the rational numbers r, 1 <= r < 2, such that the odd part of both the numerator and the denominator of r, when reduced to lowest terms, is less than or equal to the fixed odd number 2n-1. - Gene Ward Smith, Mar 27 2006
(1/4)*Number of distinct angular positions under which an observer positioned at the center of a square of a square lattice can see the (2n) X (2n) points symmetrically surrounding his position.
(1/8)*number of distinct angular positions under which an observer positioned at a lattice point of a square lattice can see the (2n+1)X(2n+1) points symmetrically surrounding his position gives A002088.
(1/2)*number of distinct angular positions under which an observer positioned at the center of an edge of a square lattice can see the (2n)X(2n-1) points symmetrically surrounding his position gives A099958.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
Lv Chuan, On the Mean Value of an Arithmetical Function, in Zhang Wenpeng (ed.), Research on Smarandache Problems in Number Theory (collected papers), 2004, pp. 89-92.
Wikipedia, Tonality diamond.
FORMULA
a(n+1) - a(n) = phi(2n+1) (A037225).
a(n) = (8/Pi^2)*n^2 + O(n^(3/2+eps)) (Lemma 1 in Lv Chuan, 2004). - Amiram Eldar, Aug 02 2022, corrected by M. F. Hasler, Mar 26 2023
MATHEMATICA
Accumulate[EulerPhi[2*Range[0, 50]+1]] (* Harvey P. Dale, Aug 20 2021 *)
PROG
(PARI) apply( {A099957(n)=sum(k=1, n, eulerphi(2*k-1))}, [1..55]) \\ M. F. Hasler, Apr 03 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Nov 13 2004
STATUS
approved