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%I #23 Jan 28 2025 12:48:13
%S 1,1,7,2,0,9,3,6,1,7,2,8,5,6,6,9,0,3,9,6,8,7,8,1,8,7,9,5,8,1,0,8,9,8,
%T 8,0,4,0,2,4,2,4,5,7,0,8,8,0,2,7,6,3,7,1,7,6,0,1,8,6,6,3,6,7,1,2,1,8,
%U 6,6,3,4,6,0,7,6,4,1,2,2,8,3,6,5,4,5,6,1,1,2,2,8,6,7,2,3,0,3,2
%N Decimal expansion of least x > 0 satisfying x^2 + 1 = tan(x).
%C 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.
%C Guide to related sequences, with graphs included in Mathematica programs:
%C a.... b.... c.... x
%C 1.... 0.... 1.... A200338
%C 1.... 0.... 2.... A200339
%C 1.... 0.... 3.... A200340
%C 1.... 0.... 4.... A200341
%C 1.... 1.... 1.... A200342
%C 1.... 1.... 2.... A200343
%C 1.... 1.... 3.... A200344
%C 1.... 1.... 4.... A200345
%C 1.... 2.... 1.... A200346
%C 1.... 2.... 2.... A200347
%C 1.... 2.... 3.... A200348
%C 1.... 2.... 4.... A200349
%C 1.... 3.... 1.... A200350
%C 1.... 3.... 2.... A200351
%C 1.... 3.... 3.... A200352
%C 1.... 3.... 4.... A200353
%C 1.... 4.... 1.... A200354
%C 1.... 4.... 2.... A200355
%C 1.... 4.... 3.... A200356
%C 1.... 4.... 4.... A200357
%C 2.... 0.... 1.... A200358
%C 2.... 0.... 3.... A200359
%C 2.... 1.... 1.... A200360
%C 2.... 1.... 2.... A200361
%C 2.... 1.... 3.... A200362
%C 2.... 1.... 4.... A200363
%C 2.... 2.... 1.... A200364
%C 2.... 2.... 3.... A200365
%C 2.... 3.... 1.... A200366
%C 2.... 3.... 2.... A200367
%C 2.... 3.... 3.... A200368
%C 2.... 3.... 4.... A200369
%C 2.... 4.... 1.... A200382
%C 2.... 4.... 3.... A200383
%C 3.... 0.... 1.... A200384
%C 3.... 0.... 2.... A200385
%C 3.... 0.... 4.... A200386
%C 3.... 1.... 1.... A200387
%C 3.... 1.... 2.... A200388
%C 3.... 1.... 3.... A200389
%C 3.... 1.... 4.... A200390
%C 3.... 2.... 1.... A200391
%C 3.... 2.... 2.... A200392
%C 3.... 2.... 3.... A200393
%C 3.... 2.... 4.... A200394
%C 3.... 3.... 1.... A200395
%C 3.... 3.... 2.... A200396
%C 3.... 3.... 4.... A200397
%C 3.... 4.... 1.... A200398
%C 3.... 4.... 2.... A200399
%C 3.... 4.... 3.... A200400
%C 3.... 4.... 4.... A200401
%C 4.... 0.... 1.... A200410
%C 4.... 0.... 3.... A200411
%C 4.... 1.... 1.... A200412
%C 4.... 1.... 2.... A200413
%C 4.... 1.... 3.... A200414
%C 4.... 1.... 4.... A200415
%C 4.... 2.... 1.... A200416
%C 4.... 2.... 3.... A200417
%C 4.... 3.... 1.... A200418
%C 4.... 3.... 2.... A200419
%C 4.... 3.... 3.... A200420
%C 4.... 3.... 4.... A200421
%C 4.... 4.... 1.... A200422
%C 4.... 4.... 3.... A200423
%C 1... -1.... 1.... A200477
%C 1... -1.... 2.... A200478
%C 1... -1.... 3.... A200479
%C 1... -1.... 4.... A200480
%C 1... -2.... 1.... A200481
%C 1... -2.... 2.... A200482
%C 1... -2.... 3.... A200483
%C 1... -2.... 4.... A200484
%C 1... -3.... 1.... A200485
%C 1... -3.... 2.... A200486
%C 1... -3.... 3.... A200487
%C 1... -3.... 4.... A200488
%C 1... -4.... 1.... A200489
%C 1... -4.... 2.... A200490
%C 1... -4.... 3.... A200491
%C 1... -4.... 4.... A200492
%C 2... -1.... 1.... A200493
%C 2... -1.... 2.... A200494
%C 2... -1.... 3.... A200495
%C 2... -1.... 4.... A200496
%C 2... -2.... 1.... A200497
%C 2... -2.... 3.... A200498
%C 2... -3.... 1.... A200499
%C 2... -3.... 2.... A200500
%C 2... -3.... 3.... A200501
%C 2... -3.... 4.... A200502
%C 2... -4.... 1.... A200584
%C 2... -4.... 3.... A200585
%C 2... -1.... 2.... A200586
%C 2... -1.... 3.... A200587
%C 2... -1.... 4.... A200588
%C 3... -2.... 1.... A200589
%C 3... -2.... 2.... A200590
%C 3... -2.... 3.... A200591
%C 3... -2.... 4.... A200592
%C 3... -3.... 1.... A200593
%C 3... -3.... 2.... A200594
%C 3... -3.... 4.... A200595
%C 3... -4.... 1.... A200596
%C 3... -4.... 2.... A200597
%C 3... -4.... 3.... A200598
%C 3... -4.... 4.... A200599
%C 4... -1.... 1.... A200600
%C 4... -1.... 2.... A200601
%C 4... -1.... 3.... A200602
%C 4... -1.... 4.... A200603
%C 4... -2.... 1.... A200604
%C 4... -2.... 3.... A200605
%C 4... -3.... 1.... A200606
%C 4... -3.... 2.... A200607
%C 4... -3.... 3.... A200608
%C 4... -3.... 4.... A200609
%C 4... -4.... 1.... A200610
%C 4... -4.... 3.... A200611
%C 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.
%C 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.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e x=1.17209361728566903968781879581089880...
%t (* Program 1: A200338 *)
%t a = 1; b = 0; c = 1;
%t f[x_] := a*x^2 + b*x + c; g[x_] := Tan[x]
%t Plot[{f[x], g[x]}, {x, -.1, Pi/2}, {AxesOrigin -> {0, 0}}]
%t r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
%t RealDigits[r] (* A200338 *)
%t (* Program 2: implicit surface of x^2+u*x+v=tan(x) *)
%t f[{x_, u_, v_}] := x^2 + u*x + v - Tan[x];
%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1.57}]}, {u, 0, 5, .1}, {v, 0, 5, .1}];
%t ListPlot3D[Flatten[t, 1]] (* for A200388 *)
%o (PARI) solve(x=1,1.2,x^2+1-tan(x)) \\ _Charles R Greathouse IV_, Mar 23 2022
%Y Cf. A197737, A198414, A198755, A198866, A199170, A199370, A199429, A199597, A199949.
%K nonn,cons
%O 1,3
%A _Clark Kimberling_, Nov 16 2011