OFFSET
1,3
COMMENTS
For many choices of a,b,c, there is exactly one x satisfying a*x^2 + b*x + c = tan(x) and 0 < x < Pi/2.
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
1.... 0.... 1.... A200338
1.... 0.... 2.... A200339
1.... 0.... 3.... A200340
1.... 0.... 4.... A200341
1.... 1.... 1.... A200342
1.... 1.... 2.... A200343
1.... 1.... 3.... A200344
1.... 1.... 4.... A200345
1.... 2.... 1.... A200346
1.... 2.... 2.... A200347
1.... 2.... 3.... A200348
1.... 2.... 4.... A200349
1.... 3.... 1.... A200350
1.... 3.... 2.... A200351
1.... 3.... 3.... A200352
1.... 3.... 4.... A200353
1.... 4.... 1.... A200354
1.... 4.... 2.... A200355
1.... 4.... 3.... A200356
1.... 4.... 4.... A200357
2.... 0.... 1.... A200358
2.... 0.... 3.... A200359
2.... 1.... 1.... A200360
2.... 1.... 2.... A200361
2.... 1.... 3.... A200362
2.... 1.... 4.... A200363
2.... 2.... 1.... A200364
2.... 2.... 3.... A200365
2.... 3.... 1.... A200366
2.... 3.... 2.... A200367
2.... 3.... 3.... A200368
2.... 3.... 4.... A200369
2.... 4.... 1.... A200382
2.... 4.... 3.... A200383
3.... 0.... 1.... A200384
3.... 0.... 2.... A200385
3.... 0.... 4.... A200386
3.... 1.... 1.... A200387
3.... 1.... 2.... A200388
3.... 1.... 3.... A200389
3.... 1.... 4.... A200390
3.... 2.... 1.... A200391
3.... 2.... 2.... A200392
3.... 2.... 3.... A200393
3.... 2.... 4.... A200394
3.... 3.... 1.... A200395
3.... 3.... 2.... A200396
3.... 3.... 4.... A200397
3.... 4.... 1.... A200398
3.... 4.... 2.... A200399
3.... 4.... 3.... A200400
3.... 4.... 4.... A200401
4.... 0.... 1.... A200410
4.... 0.... 3.... A200411
4.... 1.... 1.... A200412
4.... 1.... 2.... A200413
4.... 1.... 3.... A200414
4.... 1.... 4.... A200415
4.... 2.... 1.... A200416
4.... 2.... 3.... A200417
4.... 3.... 1.... A200418
4.... 3.... 2.... A200419
4.... 3.... 3.... A200420
4.... 3.... 4.... A200421
4.... 4.... 1.... A200422
4.... 4.... 3.... A200423
1... -1.... 1.... A200477
1... -1.... 2.... A200478
1... -1.... 3.... A200479
1... -1.... 4.... A200480
1... -2.... 1.... A200481
1... -2.... 2.... A200482
1... -2.... 3.... A200483
1... -2.... 4.... A200484
1... -3.... 1.... A200485
1... -3.... 2.... A200486
1... -3.... 3.... A200487
1... -3.... 4.... A200488
1... -4.... 1.... A200489
1... -4.... 2.... A200490
1... -4.... 3.... A200491
1... -4.... 4.... A200492
2... -1.... 1.... A200493
2... -1.... 2.... A200494
2... -1.... 3.... A200495
2... -1.... 4.... A200496
2... -2.... 1.... A200497
2... -2.... 3.... A200498
2... -3.... 1.... A200499
2... -3.... 2.... A200500
2... -3.... 3.... A200501
2... -3.... 4.... A200502
2... -4.... 1.... A200584
2... -4.... 3.... A200585
2... -1.... 2.... A200586
2... -1.... 3.... A200587
2... -1.... 4.... A200588
3... -2.... 1.... A200589
3... -2.... 2.... A200590
3... -2.... 3.... A200591
3... -2.... 4.... A200592
3... -3.... 1.... A200593
3... -3.... 2.... A200594
3... -3.... 4.... A200595
3... -4.... 1.... A200596
3... -4.... 2.... A200597
3... -4.... 3.... A200598
3... -4.... 4.... A200599
4... -1.... 1.... A200600
4... -1.... 2.... A200601
4... -1.... 3.... A200602
4... -1.... 4.... A200603
4... -2.... 1.... A200604
4... -2.... 3.... A200605
4... -3.... 1.... A200606
4... -3.... 2.... A200607
4... -3.... 3.... A200608
4... -3.... 4.... A200609
4... -4.... 1.... A200610
4... -4.... 3.... A200611
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 A200338, take f(x,u,v) = x^2 + u*x + v - tan(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.
EXAMPLE
x=1.17209361728566903968781879581089880...
MATHEMATICA
(* Program 1: A200338 *)
a = 1; b = 0; c = 1;
f[x_] := a*x^2 + b*x + c; g[x_] := Tan[x]
Plot[{f[x], g[x]}, {x, -.1, Pi/2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[r] (* A200338 *)
(* Program 2: implicit surface of x^2+u*x+v=tan(x) *)
f[{x_, u_, v_}] := x^2 + u*x + v - Tan[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1.57}]}, {u, 0, 5, .1}, {v, 0, 5, .1}];
ListPlot3D[Flatten[t, 1]] (* for A200388 *)
PROG
(PARI) solve(x=1, 1.2, x^2+1-tan(x)) \\ Charles R Greathouse IV, Mar 23 2022
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Nov 16 2011
STATUS
approved