cp's OEIS Frontend

R(n,k) is the number of ordered pairs (x,y) of integers x,y satisfying 1 <= x <= k, 1 <= y <= k, and x*y <= n.

Limiting row: A000618=(1,3,5,8,10,14,16,20,...).

Row 1: A000027

Row 2: A032766

Row 3: A106252

Row 4: A211703

Row 5: A211704

R(n,n) = A000618(n)

...

For n > =1, row n is a homogeneous linear recurrence sequence of order A005728(n), and it exemplifies a certain class, C, of recurrences which are palindromic (in the sense given below). The class depends on sequences s having n-th term [(n^k)/j], where k and j are arbitrary fixed positive integers and [ ] = floor. The characteristic polynomial of s is (x^j-1)(x-1)^k, which is a palindromic polynomial (sometimes called a reciprocal polynomial). The class C consists of sequences u given by the form

...

u(n) = c(1)*[r(1)*n^k(1)] + ... + c(m)*[r(m)*n^k(m)],

...

where c(i) are integers and r(i) are rational numbers. Assume that r(i) is in lowest terms, and let j(i) be its denominator. Then the characteristic polynomial of u is the least common multiple of all the irreducible (over the integers) factors of all the polynomials (x^j(i)-1)(x-1)^k(i). As all such factors are palindromic (indeed, they are all cyclotomic polynomials), the characteristic polynomial of u is also palindromic. In other words, if the generating function of u is written as p(x)/q(x), then q(x) is a palindromic polynomial.

Thus, if q(x) = q(h)x^h + ... + q(1)x + q(0),

then (q(h), q(h-1), ..., q(1), q(0)) is palindromic, and consequently, the recurrence coefficients for u, after excluding q(0); i.e., (- q(h-1), ... - q(1)), are palindromic. For example, row 3 of A211701 has the following recurrence: u(n) = u(n-2) + u(n-3) - u(n-5), for which q(x) = x^5 - x^3 - x^2 + 1, with recurrence coefficients (0,1,1,0,-1).

Recurrence coefficients (palindromic after excluding the last term) are shown here:

for row 1: (2, -1)

for row 2: (1 ,1, -1)

for row 3: (0, 1, 1, 0, -1)

for row 4: (0, 0, 1, 1, 0, 0, -1)

for row 5: (-1, -1, 0, 1, 2, 2, 1, 0, -1, -1, -1)

for row 6: (0, -1, 0, 0, 1, 1, 1, 1, 0, 0, -1, 0, -1)

for row 7: (-1, -2, -2, -2, -1, 0, 2, 3, 4, 4, 3, 2,

0, -1, -2, -2, -2, -1, -1)

for row 13: (-2,-4,-7,-12,-18,-27,-37,-50,-64,-80,-95,

-111,-123,-133,-137,-136,-126,-110,-84,-52,

-12,32,80,127,173,213,246,269,281,281,269,

246,213,173,127,80,32,-12,-52,-84,-110,

-126,-136,-137,-133,-123,-111,-95,-80,-64,

-50,-37,-27,-18,-12,-7,-4,-2,-1)

So, T(n, k) is also the index of fraction 1/k in the Farey fractions of order n. - Michel Marcus, Jun 27 2014

Start with array [1,k], and for each integer i from k+1 to n, insert i between every consecutive pair that sums to i. The length of the resulting array is T(n,k). For example, with n=5 and k=2 we have [1,2] -> [1,3,2] -> [1,4,3,2] -> [1,5,4,3,5,2] which has length 6, so T(5,2)=6. This is from a discovery of Leo Moser as described by Martin Gardner. - Curtis Bechtel, Oct 05 2024

The number of vertices along each edge is A005728(n). No formula for a(n) is known.

See the linked references for further details.

The first diagram where not all edge points are connected is n = 3. For example a line connecting points (0,1/3) and (1/3,0) has equation 3*y - 6*x - 1 = 0, and as one of the x or y coefficients is greater than n (3 in this case) the line is not included.