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A300291
Triangle T read by rows: T is used to obtain the denominators of all fractional values for x = cos(phi) and y = sin(phi) with (x, y) on the unit circle for 0 < phi < Pi/2.
1
5, 1, 13, 17, 1, 25, 1, 29, 1, 41, 37, 1, 1, 1, 61, 1, 53, 1, 65, 1, 85, 65, 1, 73, 1, 89, 1, 113, 1, 85, 1, 97, 1, 1, 1, 145, 101, 1, 109, 1, 1, 1, 149, 1, 181, 1, 125, 1, 137, 1, 157, 1, 185, 1, 221, 145, 1, 1, 1, 169, 1, 193, 1, 1, 1, 265, 1, 173, 1, 185, 1, 205, 1, 233, 1, 269, 1, 313, 197, 1, 205, 1, 221, 1, 1, 1, 277, 1, 317, 1, 365
OFFSET
2,1
COMMENTS
All rational values of sin(phi) = p/q and cos(phi) = r/s (gcd(p, q) = 1 = gcd(r, s), p and r nonnegative integer, q and s positive integer) with sin(phi)^2 + cos(phi)^2 = 1, and phi from [0, Pi/2] are the integer values 0 and 1, and the fractional values (q >= 2, s >= 2) with (p = p(n, m), r = r(n, m)) = (X(n, m), Y(n, m)) or (Y(n, m), X(n, m)) and q = q(n, m) = Zhat(n, m) = T(n, m), disregarding all p = 0 and r = 0 values. Here X and Y are the triangles A225950 and A225952 (with 0's) known from the legs of primitive Pythagorean triangles with odd X and even Y, respectively, and the corresponding triangle Z for the hypotenuses is A222946.
The present triangle T = Zhat is obtained from triangle Z by replacing all entries 0 (those with gcd(n, m) not 1 or (-1)^(n+m) = +1) by 1.
In each row n the positive values for X(n, m)/Zhat(n, m) decrease and for Y(n, m)/Zhat(n, m) they increase.
The maximal values of X(n, m)/Zhat(n, m) decreases from row n to row n+1 if n is even and it increases if n is odd. The maximal values of Y(n, m)/Zhat(n, m) increase from row n to row n+1.
The positive minimal values of X(n, m)/Zhat(n, m) decreases from row n to row n+1. The positive minimal values of Y(n, m)/Zhat(n, m) increase from row n to row n+1 if n is even, and they decrease if n is odd.
FORMULA
T(n, m) = 0 if n < m + 1, and T(n, m) = n^2 + m^2 if gcd(n, m) = 1 and (-1)^(n+m) = -1, and T(n, m) = 1 otherwise.
EXAMPLE
The triangle T = Zhat begins (0's are not shown):
n\m 1 2 3 4 5 6 7 8 9 10 11 ...
2: 5
3: 1 13
4: 17 1 25
5: 1 29 1 41
6: 37 1 1 1 61
7: 1 53 1 65 1 85
8: 65 1 73 1 89 1 113
9: 1 85 1 97 1 1 1 145
10: 101 1 109 1 1 1 149 1 181
11: 1 125 1 137 1 157 1 185 1 221
12: 145 1 1 1 169 1 193 1 1 1 265
...
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For the sin(phi) = p/q values from triangle X(n, m)/Zhat(n, m) = A225950(n, m)/T(n, m), and from triangle Y(n, m)/Zhat(n, m) = A225952(n, m)/T(n, m), for n = 2..20, see the attached link.
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The approximate values (3 digits) for X(n, m)/Zhat(n, m) begin:
n\m 1 2 3 4 5 6 7 8 9 ...
2: .600
3: 0 .385
4: .882 0 .280
5: 0 .724 0 .220
6: .946 0. 0 0 .180
7: 0 .849 0 .508 0 .153
8: .969 0 .753 0 .438 0 .133
9: 0 .906 0 .670 0 0 0 .117
10 .980 0 .835 0 0 0 .342 0 .105
...
The approximate values (3 digits) of Y(n, m)/Zhat(n, m) begin:
n\m 1 2 3 4 5 6 7 8 9 ...
2: .800
3: 0 .923
4: .471 0 .960
5: 0 .690 0 .976
6: .324 0 0 0 .984
7: 0 .528 0 .862 0 .988
8: .246 0 .658 0 .899 0 .99
9: 0 .424 0 .742 0 0 0 .993
10 .198 0 .550 0 0 0 .940 0 .994
...
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CROSSREFS
KEYWORD
nonn,tabl,easy
AUTHOR
Wolfdieter Lang, Mar 13 2018
STATUS
approved