A106609 -id:A106609 - cp's OEIS frontend

A367824 Array read by ascending antidiagonals: A(n, k) is the numerator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

1, 1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 1, 0, -1, -1, 1, 3, 1, -1, -3, -1, 1, 2, 1, 0, -1, -2, -1, 1, 5, 3, 1, -1, -3, -5, -1, 1, 3, 1, 1, 0, -1, -1, -3, -1, 1, 7, 5, 1, 1, -1, -1, -5, -7, -1, 1, 4, 3, 2, 1, 0, -1, -2, -3, -4, -1, 1, 0, 7, 5, 3, 1, -1, -3, -5, -7, -9, -1

Offset: 0

Stefano Spezia, Dec 02 2023

This array generalizes A367727.

			The array of the fractions begins:
  1,  -1,   -1,   -1,   -1,   -1,    -1,    -1, ...
  1,   0, -1/3, -1/2, -3/5, -2/3,  -5/7,  -3/4, ...
  1, 1/3,    0, -1/5, -1/3, -3/7,  -1/2,  -5/9, ...
  1, 1/2,  1/5,    0, -1/7, -1/4,  -1/3,  -2/5, ...
  1, 3/5,  1/3,  1/7,    0, -1/9,  -1/5, -3/11, ...
  1, 2/3,  3/7,  1/4,  1/9,    0, -1/11,  -1/6, ...
  1, 5/7,  1/2,  1/3,  1/5, 1/11,     0, -1/13, ...
  1, 3/4,  5/9,  2/5, 3/11,  1/6,  1/13,     0, ...
  ...
The array of the numerators begins:
  1, -1, -1, -1, -1, -1, -1, -1, ...
  1,  0, -1, -1, -3, -2, -5, -3, ...
  1,  1,  0, -1, -1, -3, -1, -5, ...
  1,  1,  1,  0, -1, -1, -1, -2, ...
  1,  3,  1,  1,  0, -1, -1, -3, ...
  1,  2,  3,  1,  1,  0, -1, -1, ...
  1,  5,  1,  1,  1,  1,  0, -1, ...
  1,  3,  5,  2,  3,  1,  1,  0, ...
  ...

Stefano Spezia, First 151 antidiagonals of the array

Cf. A367825 (denominator), A367826 (antidiagonal sums).

Cf. A004086, A026741, A051724, A060789, A060819, A106609, A106611, A106612, A106617, A106619, A153881 (n=0), A231190, A367727 (k=1).

A[0,0]=1; A[n_,k_]:=Numerator[(FromDigits[Reverse[IntegerDigits[n]]]-k)/(n+k)]; Table[A[n-k,k],{n,0,11},{k,0,n}]//Flatten

A(1, n) = -A026741(n-1) for n > 0.
A(2, n) = -A060819(n-2) for n > 2.
A(3, n) = -A060789(n-3) for n > 3.
A(4, n) = -A106609(n-4) for n > 3.
A(5, n) = -A106611(n-5) for n > 4.
A(6, n) = -A051724(n-6) for n > 5.
A(7, n) = -A106615(n-7) for n > 6.
A(8, n) = -A106617(n-8) = A231190(n) for n > 7.
A(9, n) = -A106619(n-9) for n > 8.
A(10, n) = -A106612(n-10) for n > 9.

A367825 Array read by ascending antidiagonals: A(n, k) is the denominator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 1, 2, 1, 1, 5, 5, 5, 5, 1, 1, 3, 3, 1, 3, 3, 1, 1, 7, 7, 7, 7, 7, 7, 1, 1, 4, 2, 4, 1, 4, 2, 4, 1, 1, 9, 9, 3, 9, 9, 3, 9, 9, 1, 10, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 1, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 4, 6, 12, 2, 3, 6, 1, 6, 3, 2, 3, 6, 1

Offset: 0

Stefano Spezia, Dec 02 2023

This array generalizes A367728.

			The array of the fractions begins:
  1,  -1,   -1,   -1,   -1,   -1,    -1,    -1, ...
  1,   0, -1/3, -1/2, -3/5, -2/3,  -5/7,  -3/4, ...
  1, 1/3,    0, -1/5, -1/3, -3/7,  -1/2,  -5/9, ...
  1, 1/2,  1/5,    0, -1/7, -1/4,  -1/3,  -2/5, ...
  1, 3/5,  1/3,  1/7,    0, -1/9,  -1/5, -3/11, ...
  1, 2/3,  3/7,  1/4,  1/9,    0, -1/11,  -1/6, ...
  1, 5/7,  1/2,  1/3,  1/5, 1/11,     0, -1/13, ...
  1, 3/4,  5/9,  2/5, 3/11,  1/6,  1/13,     0, ...
  ...
The array of the denominators begins:
  1, 1, 1, 1,  1,  1,  1,  1, ...
  1, 1, 3, 2,  5,  3,  7,  4, ...
  1, 3, 1, 5,  3,  7,  2,  9, ...
  1, 2, 5, 1,  7,  4,  3,  5, ...
  1, 5, 3, 7,  1,  9,  5, 11, ...
  1, 3, 7, 4,  9,  1, 11,  6, ...
  1, 7, 2, 3,  5, 11,  1, 13, ...
  1, 4, 9, 5, 11,  6, 13,  1, ...
  ...

Stefano Spezia, First 151 antidiagonals of the array

Cf. A367824 (numerator), A367827 (antidiagonal sums).

Cf. A000012 (n=0), A004086, A026741, A051724, A060789, A060819, A106609, A106611, A106612, A106615, A106617, A231190, A367728 (k=1).

A[0,0]=1; A[n_,k_]:=Denominator[(FromDigits[Reverse[IntegerDigits[n]]]-k)/(n+k)]; Table[A[n-k,k],{n,0,12},{k,0,n}]//Flatten

A(1, n) = A026741(n+1).
A(2, n) = A060819(n+2).
A(3, n) = A060789(n+3).
A(4, n) = A106609(n+4).
A(5, n) = A106611(n+5).
A(6, n) = A051724(n+6).
A(7, n) = A106615(n+7).
A(8, n) = A106617(n+8) = A231190(n+16).
A(9, n) = A106619(n+9).
A(10, n) = A106612(n+10).

A231189 Coefficients of the algebraic number 2sin(2Pi/n) in the power basis of Q(2*cos(Pi/q(n))), with q(n) = A225975(n), n >= 1.

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 1, 0, -3, 0, 1, 0, 0, 0, 1, 0, 5, 0, -5, 0, 1, 0, -3, 0, 1, 0, -7, 0, 14, 0, -7, 0, 1, 0, 0, 1, 0, 9, 0, -30, 0, 27, 0, -9, 0, 1, 0, 0, 0, 5, 0, -5, 0, 1, 0, -7, 0, 22, 0, -13, 0, 2, 0, -3, 0, 1, 0, 13, 0, -91, 0, 182, 0, -156, 0, 65, 0, -13, 0, 1, 0, 0, 0, -4, 0, 5, 0, -1, 0, -15, 0, 140, 0, -378, 0, 450, 0, -275, 0, 90, 0, -15, 0, 1, 0, 0, -1, 1

Offset: 1

Wolfdieter Lang, Dec 04 2013

The relevant trigonometric identity (used in the D. H. Lehmer and  I. Niven references, given in A181871) is 2*sin(2*Pi/n) = 2*cos(2*Pi*(1/n -1/4)) = 2*cos(Pi*abs(n-4)/(2*n)) = 2*cos(Pi*p(n)/q(n)), with gcd(p(n), q(n)) = 1 (fraction p(n)/q(n) in lowest terms). One finds p(n) = A106609(n-4), n >=4,  with p(1) = 3 , p(2) = 1 = p(3), and q(n) = A225975(n), n >= 1. See the comments on these two A-numbers. Therefore, 2*sin(2*Pi/n) = R(p(n), rho(q(n))), with rho(k) = 2*cos(Pi/k), and the R-polynomials (monic version of Chebyshev's T-polynomials) are given in A127672. It may happen that p(n), the degree of R, is >= delta(q(n)), the degree of the algebraic number rho(q(n)). Here delta(k) = A055034(k) is the degree of the minimal polynomial C(k, x) of rho(k) found under A187360. In this case one can reduce all rho(q(n)) powers >=  delta(q(n)) with the help of the equation C(q(n), rho(q(n))) = 0. Thus the final result is  2*sin(2*Pi/n) = R(p(n), x) (mod C(q(n), x)) with x = rho(q(n)). Because R is an integer polynomial this shows that 2*sin(2*Pi/n) is an integer in the algebraic number field Q(rho(q(n))) of degree delta(q(n)).
The power basis of  Q(rho(q(n))) is <1, rho(q(n)), ..., rho(q(n))^(delta(q(n))-1)>. Therefore the length of row n of this table is delta(q(n)).
The values n for which mod C(q(n), x) is in operation for the given formula for 2*sin(2*Pi/n) are those for which delta(q(n)) - p(n)  <= 0, that is n = 1, 2, 12, 15, 18, 20, 21, 24, 25, 27, 28, 30,...
For the minimal polynomials of 2*sin(2*Pi/n) see the coefficient table A231188.

			[0], [0], [0, 1], [2], [0, 1, 0, 0], [0, 1], [0, -3, 0, 1, 0, 0], [0, 1], [0, 5, 0, -5, 0, 1], ...
The table a(n,m) begins (the trailing zeros are needed to have the correct degree for Q(rho(q(n)))):
n\m  0   1  2   3  4    5  6    7  8    9  10  11 12  13 14 15 16 17 ...
1:   0
2:   0
3:   0   1
4:   2
5:   0   1  0   0
6:   0   1
7:   0  -3  0   1  0    0
8:   0   1
9:   0   5  0  -5  0    1
10:  0  -3  0   1
11:  0  -7  0  14  0   -7  0    1  0    0
12:  1
13:  0   9  0 -30  0   27  0   -9  0    1   0   0
14:  0   5  0  -5  0    1
15:  0  -7  0  22  0  -13  0    2
16:  0  -3  0   1
17:  0  13  0 -91  0  182  0 -156  0   65   0 -13  0   1  0  0
18:  0  -4  0   5  0   -1
19:  0 -15  0 140  0 -378  0  450  0 -275   0  90  0 -15  0  1  0  0
20: -1   1
...
--------------------------------------------------------------------------
n=1:  2*sin(2*Pi/1) = 0. rho(q(1)) = rho(2) = 2*cos(Pi/2) = 0 and p(1) = 3. R(3, x) = -3*x + x^3 and C(2, x) = x. Therefore R(3, x) (mod C(2, x)) = 0. The degree of C(2, x) is delta(2) = A055034(2) = 1. Here one should use 1 for the undefined  rho(q(1))^0 in order to obtain a(1, 0) = 0.
n=2: 2*sin(2*Pi/2) = 0; rho(q(2)) = rho(2) =  0; p(2) = 1,  R(1, x) = x , C(2, x) = x and delta(2) = 1.  Therefore   R(1, x)  (mod C(1, x)) = 0.   Again, rho(2)^0 is put to 1 here, and a(2, 0) = 0.
n=5: 2*sin(2*Pi/5) = R(1, rho(10)) (mod C(10, rho(10)) =1* rho(10) (the degree of C(10,x) is delta(10) = 4, therefore the mod prescription is not needed).  Therefore, a(5, 0) =0, a(5,1) =1, a(n, m) = 0 for m=2, 3.
n =11: 2*sin(2*Pi/11) = R(7, x) (mod(C(22, x)) with x = rho(22), because p(11) = 7 and q(11) = 22. The degree of C(22, x) is delta(22) = 10, therefore the mod restriction is not needed and R(7, x) = -7*x + 14*x^3 - 7*x^5 + x^7. The coefficients produce the row [0, -7, 0, 14,  0,  -7, 0, 1, 0, 0] with the two trailing zeros needed to obtain the correct row length, namely delta(q(11)) = 10.

Cf. A055034 (for delta), A106609 (for p), A225975 (for q), A127672 (for R), A187360 (for C), A181871, A231188.

a(n,m) = [x^m] (R(p(n), x) (mod C(q(n), x)), n >= 1, m = 0, 1, ..., delta(q(n)) - 1, where the R and C polynomials are found in A187360 and A127672, respectively. p(n) = A106609(n-4), n >=4, with p(1) = 3 , p(2) = 1 = p(3), and q(n) = A225975(n). Powers of x = rho(q(n)) = 2*cos(Pi/q(n)) appear in the table in increasing order.

cp's OEIS Frontend

A367824 Array read by ascending antidiagonals: A(n, k) is the numerator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

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A367825 Array read by ascending antidiagonals: A(n, k) is the denominator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

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A231189 Coefficients of the algebraic number 2sin(2Pi/n) in the power basis of Q(2*cos(Pi/q(n))), with q(n) = A225975(n), n >= 1.

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cp's OEIS Frontend

A367824 Array read by ascending antidiagonals: A(n, k) is the numerator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

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Programs

Mathematica

Formula

A367825 Array read by ascending antidiagonals: A(n, k) is the denominator of (R(n) - k)/(n + k), where R(n) is the digit reversal of n, with A(0, 0) = 1.

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Programs

Mathematica

Formula

A231189 Coefficients of the algebraic number 2*sin(2*Pi/n) in the power basis of Q(2*cos(Pi/q(n))), with q(n) = A225975(n), n >= 1.

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Comments

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Formula

A231189 Coefficients of the algebraic number 2sin(2Pi/n) in the power basis of Q(2*cos(Pi/q(n))), with q(n) = A225975(n), n >= 1.