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A082375 Irregular triangle read by rows: row n begins with n and decreases by 2 until 0 or 1 is reached, for n >= 0. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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) see the example section. (End)

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..1000

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

EXAMPLE

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

... reformatted by Wolfdieter Lang, Feb 15 2020

From Wolfdieter Lang, Feb 17 2020: (Start)

Conversion identity:

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)

MATHEMATICA

Flatten[Table[Range[n, 0, -2], {n, 0, 20}]] (* Harvey P. Dale, Apr 03 2019 *)

PROG

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

CROSSREFS

Cf. A008619, A122197, A199474.

Sequence in context: A194549 A063277 A029178 * A025644 A022332 A215589

Adjacent sequences:  A082372 A082373 A082374 * A082376 A082377 A082378

KEYWORD

nonn,tabf

AUTHOR

Michael Somos, Apr 09 2003

STATUS

approved

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Last modified August 10 20:21 EDT 2022. Contains 356039 sequences. (Running on oeis4.)