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A211701 Rectangular array by antidiagonals, n>=1, k>=1: R(n,k)=n+[n/2]+...+[n/k], where [ ]=floor. 13
1, 2, 1, 3, 3, 1, 4, 4, 3, 1, 5, 6, 5, 3, 1, 6, 7, 7, 5, 3, 1, 7, 9, 8, 8, 5, 3, 1, 8, 10, 11, 9, 8, 5, 3, 1, 9, 12, 12, 12, 10, 8, 5, 3, 1, 10, 13, 14, 13, 13, 10, 8, 5, 3, 1, 11, 15, 16, 16, 14, 14, 10, 8, 5, 3, 1, 12, 16, 18, 18, 17, 15, 14, 10, 8, 5, 3, 1, 13, 18, 19, 20, 19 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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)

LINKS

Table of n, a(n) for n=1..83.

EXAMPLE

Northwest corner:

1...2...3...4...5...6....7....8....9....10

1...3...4...6...7...9....10...12...13...15

1...3...5...7...8...11...12...14...16...18

1...3...5...8...9...12...13...16...18...19

MATHEMATICA

f[n_, m_] := Sum[Floor[n/k], {k, 1, m}]

TableForm[Table[f[n, m], {m, 1, 20}, {n, 1, 16}]]

Flatten[Table[f[n + 1 - m, m], {n, 1, 14}, {m, 1, n}]]

CROSSREFS

Cf. A211702, A211703, A211704, A211705.

Sequence in context: A210258 A181108 A211782 * A183110 A117895 A188002

Adjacent sequences:  A211698 A211699 A211700 * A211702 A211703 A211704

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Apr 19 2012

STATUS

approved

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Last modified April 25 00:46 EDT 2017. Contains 285346 sequences.