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%I #21 Apr 07 2020 23:20:27
%S 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,
%T 113,1,85,1,97,1,1,1,145,101,1,109,1,1,1,149,1,181,1,125,1,137,1,157,
%U 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
%N 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.
%C 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.
%C 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.
%C 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.
%C 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.
%C 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.
%H Wolfdieter Lang, <a href="/A300291/a300291_2.pdf">Triangles X/Zhat and Y/Zhat for fractional values of sine (or cosine) for the unit circle, for n = 2..20.</a>
%F 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.
%e The triangle T = Zhat begins (0's are not shown):
%e n\m 1 2 3 4 5 6 7 8 9 10 11 ...
%e 2: 5
%e 3: 1 13
%e 4: 17 1 25
%e 5: 1 29 1 41
%e 6: 37 1 1 1 61
%e 7: 1 53 1 65 1 85
%e 8: 65 1 73 1 89 1 113
%e 9: 1 85 1 97 1 1 1 145
%e 10: 101 1 109 1 1 1 149 1 181
%e 11: 1 125 1 137 1 157 1 185 1 221
%e 12: 145 1 1 1 169 1 193 1 1 1 265
%e ...
%e -----------------------------------------------------------------------------
%e 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.
%e -----------------------------------------------------------------------------
%e The approximate values (3 digits) for X(n, m)/Zhat(n, m) begin:
%e n\m 1 2 3 4 5 6 7 8 9 ...
%e 2: .600
%e 3: 0 .385
%e 4: .882 0 .280
%e 5: 0 .724 0 .220
%e 6: .946 0. 0 0 .180
%e 7: 0 .849 0 .508 0 .153
%e 8: .969 0 .753 0 .438 0 .133
%e 9: 0 .906 0 .670 0 0 0 .117
%e 10 .980 0 .835 0 0 0 .342 0 .105
%e ...
%e The approximate values (3 digits) of Y(n, m)/Zhat(n, m) begin:
%e n\m 1 2 3 4 5 6 7 8 9 ...
%e 2: .800
%e 3: 0 .923
%e 4: .471 0 .960
%e 5: 0 .690 0 .976
%e 6: .324 0 0 0 .984
%e 7: 0 .528 0 .862 0 .988
%e 8: .246 0 .658 0 .899 0 .99
%e 9: 0 .424 0 .742 0 0 0 .993
%e 10 .198 0 .550 0 0 0 .940 0 .994
%e ...
%e -----------------------------------------------------------------------------
%Y Cf. A222946, A225950, A225952.
%K nonn,tabl,easy
%O 2,1
%A _Wolfdieter Lang_, Mar 13 2018