A082375 Irregular triangle read by rows: row n begins with n and decreases by 2 until 0 or 1 is reached, for n >= 0.

0, 1, 2, 0, 3, 1, 4, 2, 0, 5, 3, 1, 6, 4, 2, 0, 7, 5, 3, 1, 8, 6, 4, 2, 0, 9, 7, 5, 3, 1, 10, 8, 6, 4, 2, 0, 11, 9, 7, 5, 3, 1, 12, 10, 8, 6, 4, 2, 0, 13, 11, 9, 7, 5, 3, 1, 14, 12, 10, 8, 6, 4, 2, 0, 15, 13, 11, 9, 7, 5, 3, 1, 16, 14, 12, 10, 8, 6, 4, 2, 0, 17, 15, 13, 11, 9, 7, 5, 3, 1, 18, 16, 14

Offset: 0

Author

Michael Somos, Apr 09 2003

Keywords

• nonn
• tabf

Comments

As a sequence, a(n) = A025644(n+1) for n <= 142.

The length of row n is given by A008619(n) = 1 + floor(n/2).

From Wolfdieter Lang, Feb 17 2020: (Start)

This table T(n, m) can be used for the conversion identity

2*cos(Pi*k/N) = 2*sin((Pi/(2*N))*(N - 2*k)) = 2*sin((Pi/(2*N))*T(N-2, k-1)), here for N = n+2 >= 2, and k = m + 1 = 1, 2, ..., floor(N/2).

2*cos((Pi/N)*k) = R(k, rho(N)), where R is a monic Chebyshev polynomial from A127672 and rho(N) = 2*cos(Pi/N), gives part of the roots of the polynomial S(N-1, x), for k = 1, 2, ..., floor(N/2), with the Chebyshev S polynomials from A049310.

2*sin((Pi/(2*N))*q) = d^{(2*N)}_q/r, for q = 1, 2, ..., N, with the length ratio (q-th diagonal)/r, where r is the radius of the circle circumscribing a regular (2*N)-gon. The counting q starts with the diagonal d^{(2*N)}_1 = s(2*N) (in units of r), the side of the (2*N)-gon. The next diagonal is d^{(2*N)}_2 = rho(2*N)*s(2*N) (in units of r).

For the instances N = 4 (n = 2) and 5 (n = 3), we have:

N = n+2 = 4:

k = m+1 = 1, 2*cos(Pi*1/4) = 2*sin(Pi*2/8) = sqrt(2);

k = 2, 2*cos(Pi*2/4) = 2*sin(Pi*0/8) = 0.

N = 5 (n=3):

k=1 (m=0), 2*cos(Pi*1/5) = 2*sin(Pi*3/10) = (1 + sqrt(5))/2 = rho(5) = A001622;

k=2: 2*cos(Pi*2/5) = 2*sin(Pi*1/10) = rho(5) - 1. (End)

If b > 0 and c > 0 are the integer coefficients of a monic quadratic x^2 + b*x + c, it has integer roots if its discriminant d^2 = b^2 - 4c is a perfect square. This sequence is the values of d for increasing b sorted by b then c. The first pair of (b, c) = (2, 1) and has d = a(0) = 0. The n-th pair of (b, c) = (A027434(n), A350634(n)) and has d = a(n-1). - Frank M Jackson, Jan 20 2024

This sequence is related to an instance of Clark Kimberling's generic dispersion arrays; in this case the leader sequence is the square numbers A000290 (without 0), and the follower sequence is the nonsquare numbers A000037. This sequence gives the 0-origin column index of n in the resulting dispersion array. - Allan C. Wechsler, Feb 26 2025

Examples

			The irregular triangle T(n, m) begins:
  n\m  0 1 2 3 4 5 ...
  0:   0
  1:   1
  2:   2 0
  3:   3 1
  4:   4 2 0
  5:   5 3 1
  6:   6 4 2 0
  7:   7 5 3 1
  8:   8 6 4 2 0
  9:   9 7 5 3 1
  10: 10 8 6 4 2 0
  ...
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Clark Kimberling, Interspersions and Dispersions.
• Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.

Crossrefs

Cf. A025644, A008619, A008619, A027434, A122197, A199474, A350634.

Programs

Mathematica

Flatten[Table[Range[n,0,-2],{n,0,20}]] (* Harvey P. Dale, Apr 03 2019 *)
lst = {}; Do[If[IntegerQ[d=Sqrt[b^2-4c]], AppendTo[lst, d]], {b, 1, 20}, {c, 1, b^2/4}]; lst (* Frank M Jackson, Jan 20 2024 *)

PARI

a(n)=local(m); if(n<0,0,m=sqrtint(1+4*n); m-1-(1+4*n-m^2)\2)

Formula

T(n, m) = n - 2*m, m = 0, 1, ..., floor(n/2), n >= 0 (see the name and programs). - Wolfdieter Lang, Feb 17 2020

a(n) = A199474(n+1) - A122197(n+1). - Wesley Ivan Hurt, Jan 09 2022

a(n) = sqrt((A027434(n+1))^2 - 4*A350634(n+1)). - Frank M Jackson, Jan 20 2024