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A201397 Decimal expansion of x satisfying x^2 + 2 = sec(x) and 0 < x < Pi. 46
1, 2, 9, 5, 4, 5, 9, 6, 4, 6, 4, 1, 5, 4, 7, 8, 7, 6, 8, 6, 2, 9, 9, 1, 3, 2, 7, 0, 7, 1, 8, 6, 4, 1, 5, 8, 9, 7, 6, 7, 2, 7, 4, 8, 2, 7, 0, 6, 8, 7, 1, 3, 1, 6, 1, 6, 0, 5, 1, 8, 1, 4, 3, 0, 2, 1, 7, 4, 9, 5, 1, 2, 6, 5, 9, 9, 3, 0, 9, 5, 5, 9, 7, 8, 6, 7, 4, 3, 9, 4, 7, 1, 9, 8, 8, 4, 7, 9, 9 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
For many choices of a and c, there are exactly two values of x satisfying a*x^2 + c = sec(x) and 0 < x < Pi. Guide to related sequences, with graphs included in Mathematica programs:
a.... c.... x
1.... 1.... A196816
1.... 2.... A201397
1.... 3.... A201398
1.... 4.... A201399
1.... 5.... A201400
1.... 6.... A201401
1.... 7.... A201402
1.... 8.... A201403
1.... 9.... A201404
1.... 10... A201405
2.... 0.... A201406, A201407
3.... 0.... A201408, A201409
4.... 0.... A201410, A201411
5.... 0.... A201412, A201413
6.... 0.... A201414, A201415
7.... 0.... A201416, A201417
8.... 0.... A201418, A201419
9.... 0.... A201420, A201421
10... 0.... A201422, A201423
3... -1.... A201515, A201516
4... -1.... A201517, A201518
5... -1.... A201519, A201520
6... -1.... A201521, A201522
7... -1.... A201523, A201524
8... -1.... A201525, A201526
9... -1.... A201527, A201528
10.. -1.... A201529, A201530
2.... 3.... A201531
3.... 2.... A200619
Suppose that f(x,u,v) is a function of three real variables and that g(u,v) is a function defined implicitly by f(g(u,v),u,v)=0. We call the graph of z=g(u,v) an implicit surface of f.
For an example related to A201397, take f(x,u,v) = u*x^2 + v = sec(x) and g(u,v) = a nonzero solution x of f(x,u,v)=0. If there is more than one nonzero solution, care must be taken to ensure that the resulting function g(u,v) is single-valued and continuous. A portion of an implicit surface is plotted by Program 2 in the Mathematica section.
LINKS
EXAMPLE
x=1.2954596464154787686299132707186415897672...
MATHEMATICA
(* Program 1: A201397 *)
a = 1; c = 2;
f[x_] := a*x^2 + c; g[x_] := Sec[x]
Plot[{f[x], g[x]}, {x, 0, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r] (* A201397 *)
(* Program 2: implicit surface of u*x^2+v=sec(x) *)
Remove["Global`*"];
f[{x_, u_, v_}] := u*x^2 + v - Sec[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .1, 1}]}, {v, 0, 1}, {u, 2 + v, 10}];
ListPlot3D[Flatten[t, 1]] (* for A201397 *)
CROSSREFS
Sequence in context: A021776 A241995 A019708 * A077124 A051491 A339204
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Dec 01 2011
STATUS
approved

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Last modified December 3 18:21 EST 2023. Contains 367540 sequences. (Running on oeis4.)