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 A200338 Decimal expansion of least x > 0 satisfying x^2 + 1 = tan(x). 159
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 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. LINKS Table of n, a(n) for n=1..99. 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 Cf. A197737, A198414, A198755, A198866, A199170, A199370, A199429, A199597, A199949. Sequence in context: A093954 A177703 A266814 * A351480 A153589 A340617 Adjacent sequences: A200335 A200336 A200337 * A200339 A200340 A200341 KEYWORD nonn,cons AUTHOR Clark Kimberling, Nov 16 2011 STATUS approved

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Last modified September 11 17:54 EDT 2024. Contains 375839 sequences. (Running on oeis4.)