A199597 -id:A199597 - cp's OEIS frontend

A200338 Decimal expansion of least x > 0 satisfying x^2 + 1 = tan(x).

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

Offset: 1

Clark Kimberling, Nov 16 2011

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
... 0.... 1.... A200338
... 0.... 2.... A200339
... 0.... 3.... A200340
... 0.... 4.... A200341
... 1.... 1.... A200342
... 1.... 2.... A200343
... 1.... 3.... A200344
... 1.... 4.... A200345
... 2.... 1.... A200346
... 2.... 2.... A200347
... 2.... 3.... A200348
... 2.... 4.... A200349
... 3.... 1.... A200350
... 3.... 2.... A200351
... 3.... 3.... A200352
... 3.... 4.... A200353
... 4.... 1.... A200354
... 4.... 2.... A200355
... 4.... 3.... A200356
... 4.... 4.... A200357
... 0.... 1.... A200358
... 0.... 3.... A200359
... 1.... 1.... A200360
... 1.... 2.... A200361
... 1.... 3.... A200362
... 1.... 4.... A200363
... 2.... 1.... A200364
... 2.... 3.... A200365
... 3.... 1.... A200366
... 3.... 2.... A200367
... 3.... 3.... A200368
... 3.... 4.... A200369
... 4.... 1.... A200382
... 4.... 3.... A200383
... 0.... 1.... A200384
... 0.... 2.... A200385
... 0.... 4.... A200386
... 1.... 1.... A200387
... 1.... 2.... A200388
... 1.... 3.... A200389
... 1.... 4.... A200390
... 2.... 1.... A200391
... 2.... 2.... A200392
... 2.... 3.... A200393
... 2.... 4.... A200394
... 3.... 1.... A200395
... 3.... 2.... A200396
... 3.... 4.... A200397
... 4.... 1.... A200398
... 4.... 2.... A200399
... 4.... 3.... A200400
... 4.... 4.... A200401
... 0.... 1.... A200410
... 0.... 3.... A200411
... 1.... 1.... A200412
... 1.... 2.... A200413
... 1.... 3.... A200414
... 1.... 4.... A200415
... 2.... 1.... A200416
... 2.... 3.... A200417
... 3.... 1.... A200418
... 3.... 2.... A200419
... 3.... 3.... A200420
... 3.... 4.... A200421
... 4.... 1.... A200422
... 4.... 3.... A200423
.. -1.... 1.... A200477
.. -1.... 2.... A200478
.. -1.... 3.... A200479
.. -1.... 4.... A200480
.. -2.... 1.... A200481
.. -2.... 2.... A200482
.. -2.... 3.... A200483
.. -2.... 4.... A200484
.. -3.... 1.... A200485
.. -3.... 2.... A200486
.. -3.... 3.... A200487
.. -3.... 4.... A200488
.. -4.... 1.... A200489
.. -4.... 2.... A200490
.. -4.... 3.... A200491
.. -4.... 4.... A200492
.. -1.... 1.... A200493
.. -1.... 2.... A200494
.. -1.... 3.... A200495
.. -1.... 4.... A200496
.. -2.... 1.... A200497
.. -2.... 3.... A200498
.. -3.... 1.... A200499
.. -3.... 2.... A200500
.. -3.... 3.... A200501
.. -3.... 4.... A200502
.. -4.... 1.... A200584
.. -4.... 3.... A200585
.. -1.... 2.... A200586
.. -1.... 3.... A200587
.. -1.... 4.... A200588
.. -2.... 1.... A200589
.. -2.... 2.... A200590
.. -2.... 3.... A200591
.. -2.... 4.... A200592
.. -3.... 1.... A200593
.. -3.... 2.... A200594
.. -3.... 4.... A200595
.. -4.... 1.... A200596
.. -4.... 2.... A200597
.. -4.... 3.... A200598
.. -4.... 4.... A200599
.. -1.... 1.... A200600
.. -1.... 2.... A200601
.. -1.... 3.... A200602
.. -1.... 4.... A200603
.. -2.... 1.... A200604
.. -2.... 3.... A200605
.. -3.... 1.... A200606
.. -3.... 2.... A200607
.. -3.... 3.... A200608
.. -3.... 4.... A200609
.. -4.... 1.... A200610
.. -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.

			x=1.17209361728566903968781879581089880...

Index entries for transcendental numbers.

Cf. A197737, A198414, A198755, A198866, A199170, A199370, A199429, A199597, A199949.

(* 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 *)

solve(x=1,1.2,x^2+1-tan(x)) \\ Charles R Greathouse IV, Mar 23 2022

A201946 Decimal expansion of x>0 satisfying x*sinh(x)=2.

Original entry on oeis.org

1, 2, 4, 9, 3, 9, 4, 3, 3, 6, 6, 4, 6, 3, 2, 4, 4, 7, 2, 5, 1, 1, 2, 7, 4, 3, 2, 1, 2, 6, 1, 0, 0, 8, 1, 2, 3, 4, 6, 9, 4, 4, 1, 3, 0, 0, 9, 0, 1, 5, 2, 9, 6, 9, 6, 2, 9, 7, 2, 6, 0, 7, 6, 8, 6, 8, 8, 2, 1, 2, 3, 9, 7, 3, 1, 0, 6, 2, 2, 9, 8, 3, 0, 0, 5, 3, 2, 5, 7, 7, 0, 8, 2, 8, 6, 5, 3, 9, 1

Offset: 1

Clark Kimberling, Dec 15 2011

For many choices of u and v, there is exactly one x>0 satisfying x*sinh(u*x)=v.  Guide to related sequences, with graphs included in Mathematica programs:
u.... v.... x
1.... 1.... A133867
1.... 2.... A201946
1.... 3.... A202243
2.... 1.... A202244
3.... 1.... A202245
2.... 2.... A202284
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 A199597, take f(x,u,v)=x*sinh(ux)-v 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.

			1.2493943366463244725112743212610081234694...

Index entries for transcendental numbers.

Cf. A201939.

(* Program 1:  A201946 *)
u = 1; v = 2;
f[x_] := x*Sinh[u*x]; g[x_] := v
Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r]   (* A201946 *)
(* Program 2: implicit surface of u*sinh(x)=v *)
f[{x_, u_, v_}] := x*Sinh[u*x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .2}]}, {v, 0, 10}, {u, 1, 4}];
ListPlot3D[Flatten[t, 1]] (* for A201946 *)

A199603 Decimal expansion of least x satisfying x+3*cos(x)=0.

Original entry on oeis.org

1, 1, 7, 0, 1, 2, 0, 9, 5, 0, 0, 0, 2, 6, 2, 6, 0, 5, 3, 7, 0, 6, 0, 4, 3, 0, 1, 1, 8, 5, 8, 9, 7, 1, 0, 8, 1, 9, 9, 9, 4, 0, 2, 0, 2, 5, 9, 5, 5, 5, 1, 5, 6, 4, 7, 4, 1, 6, 8, 1, 9, 7, 4, 4, 3, 7, 0, 0, 2, 4, 9, 2, 4, 7, 7, 1, 7, 3, 0, 8, 4, 2, 6, 4, 7, 9, 9, 0, 8, 0, 1, 1, 0, 0, 7, 0, 8, 0, 5

Offset: 1

Clark Kimberling, Nov 08 2011

See A199597 for a guide to related sequences. The Mathematica program includes a graph.

			least: -1.1701209500026260537060430118589710...
greatest:  2.9381003939708118076581364784259...

Index entries for transcendental numbers.

Cf. A199597.

a = 1; b = 3; c = 0;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -1.5, 3.5}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1.2, -1.1}, WorkingPrecision -> 110]
RealDigits[r]  (* A199603 least of 4 roots *)
r = x /. FindRoot[f[x] == g[x], {x, 2.93, 2.94}, WorkingPrecision -> 110]
RealDigits[r]  (* A199604 greatest of 4 roots *)

A199604 Decimal expansion of greatest x satisfying x+3*cos(x) = 0.

Original entry on oeis.org

2, 9, 3, 8, 1, 0, 0, 3, 9, 3, 9, 7, 0, 8, 1, 1, 8, 0, 7, 6, 5, 8, 1, 3, 6, 4, 7, 8, 4, 2, 5, 9, 1, 2, 9, 5, 9, 6, 7, 0, 2, 1, 8, 6, 1, 7, 3, 2, 2, 3, 1, 0, 1, 7, 8, 4, 6, 7, 1, 7, 6, 3, 8, 5, 3, 5, 4, 6, 7, 8, 5, 9, 2, 9, 2, 8, 3, 6, 7, 4, 6, 4, 2, 0, 8, 7, 7, 5, 5, 2, 1, 0, 3, 9, 6, 7, 7, 7, 3, 9

Offset: 1

Clark Kimberling, Nov 08 2011

See A199597 for a guide to related sequences. The Mathematica program includes a graph.

			least: -1.1701209500026260537060430118589710...
greatest:  2.9381003939708118076581364784259...

Index entries for transcendental numbers.

Cf. A199597.

a = 1; b = 3; c = 0;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -1.5, 3.5}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1.2, -1.1}, WorkingPrecision -> 110]
RealDigits[r]  (* A199603 least of 4 roots *)
r = x /. FindRoot[f[x] == g[x], {x, 2.93, 2.94}, WorkingPrecision -> 110]
RealDigits[r]  (* A199604 greatest of 4 roots *)

a(86) onwards corrected by Georg Fischer, Aug 03 2021

A199605 Decimal expansion of least x satisfying x^2+3xcos(x)=sin(x).

Original entry on oeis.org

9, 3, 0, 4, 9, 5, 0, 0, 2, 6, 3, 5, 9, 7, 0, 1, 0, 9, 7, 6, 3, 3, 4, 1, 0, 2, 4, 0, 2, 5, 4, 7, 8, 5, 1, 2, 5, 8, 6, 4, 4, 8, 8, 2, 4, 4, 1, 4, 8, 8, 7, 7, 2, 8, 1, 5, 2, 2, 7, 2, 8, 2, 8, 2, 0, 8, 3, 0, 6, 9, 2, 4, 0, 1, 9, 7, 5, 5, 2, 4, 2, 5, 1, 0, 1, 9, 2, 8, 9, 1, 3, 1, 7, 8, 3, 6, 8, 9, 9

Offset: 0

Clark Kimberling, Nov 08 2011

See A199597 for a guide to related sequences. The Mathematica program includes a graph.

			least: -0.93049500263597010976334102402547851258644...
greatest:  3.01796308106862887266781443388576897037832...

Index entries for transcendental numbers.

Cf. A199597.

a = 1; b = 3; c = 1;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -1.5, 3.5}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1, -.9}, WorkingPrecision -> 110]
RealDigits[r]  (* A199605, least of 4 roots *)
r = x /. FindRoot[f[x] == g[x], {x, 3, 3.1}, WorkingPrecision -> 110]
RealDigits[r]  (* A199606, greatest of 4 roots *)

A199606 Decimal expansion of greatest x satisfying x^2+3xcos(x)=sin(x).

Original entry on oeis.org

3, 0, 1, 7, 9, 6, 3, 0, 8, 1, 0, 6, 8, 6, 2, 8, 8, 7, 2, 6, 6, 7, 8, 1, 4, 4, 3, 3, 8, 8, 5, 7, 6, 8, 9, 7, 0, 3, 7, 8, 3, 2, 7, 2, 9, 4, 7, 3, 8, 3, 3, 3, 0, 9, 4, 0, 6, 2, 7, 6, 8, 4, 4, 5, 7, 5, 7, 0, 0, 2, 3, 7, 8, 0, 9, 9, 2, 1, 2, 9, 5, 1, 4, 6, 0, 3, 3, 7, 8, 7, 6, 8, 4, 3, 4, 7, 5, 0, 7

Offset: 1

Clark Kimberling, Nov 08 2011

See A199597 for a guide to related sequences. The Mathematica program includes a graph.

			least: -0.93049500263597010976334102402547851258644...
greatest:  3.01796308106862887266781443388576897037832...

Index entries for transcendental numbers.

Cf. A199597.

a = 1; b = 3; c = 1;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -1.5, 3.5}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1, -.9}, WorkingPrecision -> 110]
RealDigits[r]  (* A199605, least of 4 roots *)
r = x /. FindRoot[f[x] == g[x], {x, 3, 3.1}, WorkingPrecision -> 110]
RealDigits[r]  (* A199606, greatest of 4 roots *)

A199607 Decimal expansion of least x satisfying x^2+3xcos(x)=2*sin(x).

Original entry on oeis.org

5, 9, 7, 3, 3, 9, 2, 5, 0, 3, 6, 4, 8, 5, 3, 9, 7, 5, 0, 0, 4, 9, 7, 3, 6, 1, 3, 5, 9, 9, 7, 6, 6, 9, 0, 2, 8, 3, 3, 1, 8, 5, 7, 5, 6, 4, 1, 8, 4, 9, 2, 4, 1, 1, 3, 2, 7, 4, 2, 3, 8, 5, 1, 2, 2, 2, 8, 8, 6, 9, 5, 9, 3, 7, 4, 7, 8, 7, 0, 0, 7, 9, 2, 5, 4, 4, 7, 4, 1, 3, 0, 9, 1, 3, 3, 4, 4, 3, 4

Offset: 0

Clark Kimberling, Nov 08 2011

See A199597 for a guide to related sequences. The Mathematica program includes a graph.

			least: -0.5973392503648539750049736135997669028331...
greatest:  3.0481385953651166891446050593739052208...

Index entries for transcendental numbers.

Cf. A199597.

a = 1; b = 3; c = 2;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -2, 4}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -.6, -.5}, WorkingPrecision -> 110]
RealDigits[r]  (* A199607, least of 4 roots *)
r = x /. FindRoot[f[x] == g[x], {x, 3, 3.1}, WorkingPrecision -> 110]
RealDigits[r]  (* A199708, greatest of 4 roots *)

A199609 Decimal expansion of least x>0 satisfying x^2+3xcos(x)=3*sin(x).

Original entry on oeis.org

1, 1, 4, 2, 2, 5, 6, 4, 0, 2, 2, 4, 4, 7, 4, 0, 1, 1, 0, 0, 4, 4, 6, 1, 5, 8, 7, 8, 2, 3, 5, 8, 6, 4, 3, 5, 2, 5, 1, 5, 3, 4, 4, 8, 3, 4, 4, 5, 7, 6, 4, 5, 7, 4, 8, 1, 0, 1, 7, 4, 4, 4, 6, 2, 4, 3, 1, 6, 6, 5, 1, 6, 7, 4, 3, 3, 7, 0, 9, 4, 5, 1, 6, 0, 9, 7, 2, 6, 6, 3, 4, 9, 3, 4, 7, 6, 2, 6, 6

Offset: 1

Clark Kimberling, Nov 08 2011

See A199597 for a guide to related sequences. The Mathematica program includes a graph.

			least: 1.14225640224474011004461587823586435251534483...
greatest:  3.0656207603368585618674575528508213250654...

Index entries for transcendental numbers.

Cf. A199597.

a = 1; b = 3; c = 3;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -1, 4}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[r]  (* A199609, least x>0 of 3 roots *)
r = x /. FindRoot[f[x] == g[x], {x, 3, 3.1}, WorkingPrecision -> 110]
RealDigits[r]  (* A199610, greatest of 3 roots *)

A199610 Decimal expansion of greatest x satisfying x^2+3xcos(x)=3*sin(x).

Original entry on oeis.org

3, 0, 6, 5, 6, 2, 0, 7, 6, 0, 3, 3, 6, 8, 5, 8, 5, 6, 1, 8, 6, 7, 4, 5, 7, 5, 5, 2, 8, 5, 0, 8, 2, 1, 3, 2, 5, 0, 6, 5, 4, 0, 2, 0, 6, 8, 2, 0, 1, 7, 0, 6, 2, 6, 3, 9, 9, 4, 5, 6, 9, 0, 5, 4, 3, 3, 1, 2, 5, 4, 8, 2, 7, 3, 8, 3, 4, 7, 4, 3, 0, 4, 4, 5, 7, 0, 8, 1, 7, 8, 0, 0, 8, 7, 6, 1, 4, 1, 1

Offset: 1

Clark Kimberling, Nov 08 2011

See A199597 for a guide to related sequences. The Mathematica program includes a graph.

			least: 1.14225640224474011004461587823586435251534483...
greatest:  3.0656207603368585618674575528508213250654...

Index entries for transcendental numbers.

Cf. A199597.

a = 1; b = 3; c = 3;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -1, 4}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[r]  (* A199609, least of 3 roots *)
r = x /. FindRoot[f[x] == g[x], {x, 3, 3.1}, WorkingPrecision -> 110]
RealDigits[r]  (* A199610, greatest of 3 roots *)

A199611 Decimal expansion of least x satisfying x+4*cos(x)=0.

Original entry on oeis.org

1, 2, 5, 2, 3, 5, 3, 2, 3, 4, 0, 0, 2, 5, 8, 8, 7, 6, 3, 1, 8, 6, 3, 2, 8, 1, 2, 1, 9, 7, 5, 3, 8, 0, 4, 3, 5, 9, 0, 1, 2, 8, 0, 6, 1, 0, 5, 6, 6, 1, 8, 9, 9, 9, 2, 3, 8, 6, 1, 4, 4, 3, 1, 3, 0, 8, 0, 8, 0, 2, 4, 1, 3, 3, 5, 3, 2, 6, 7, 5, 6, 7, 8, 9, 0, 9, 6, 2, 7, 6, 9, 1, 9, 2, 7, 6, 2, 0, 1

Offset: 1

Clark Kimberling, Nov 08 2011

See A199597 for a guide to related sequences. The Mathematica program includes a graph.

			least: -1.25235323400258876318632812197538043590128...
greatest:  3.59530486716154799187760693508341871491...

Index entries for transcendental numbers.

Cf. A199597.

a = 1; b = 4; c = 0;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -2, 4}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1.3, -1.2}, WorkingPrecision -> 110]
RealDigits[r]  (* A199611, least of 4 roots *)
r = x /. FindRoot[f[x] == g[x], {x, 3.5, 3.6}, WorkingPrecision -> 110]
RealDigits[r]  (* A199612, greatest of 4 roots *)

cp's OEIS Frontend

A200338 Decimal expansion of least x > 0 satisfying x^2 + 1 = tan(x).

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A201946 Decimal expansion of x>0 satisfying x*sinh(x)=2.

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A199603 Decimal expansion of least x satisfying x+3*cos(x)=0.

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A199604 Decimal expansion of greatest x satisfying x+3*cos(x) = 0.

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A199605 Decimal expansion of least x satisfying x^2+3*x*cos(x)=sin(x).

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A199606 Decimal expansion of greatest x satisfying x^2+3*x*cos(x)=sin(x).

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Mathematica

A199607 Decimal expansion of least x satisfying x^2+3*x*cos(x)=2*sin(x).

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Mathematica

A199609 Decimal expansion of least x>0 satisfying x^2+3*x*cos(x)=3*sin(x).

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A199605 Decimal expansion of least x satisfying x^2+3xcos(x)=sin(x).

A199606 Decimal expansion of greatest x satisfying x^2+3xcos(x)=sin(x).

A199607 Decimal expansion of least x satisfying x^2+3xcos(x)=2*sin(x).

A199609 Decimal expansion of least x>0 satisfying x^2+3xcos(x)=3*sin(x).

A199610 Decimal expansion of greatest x satisfying x^2+3xcos(x)=3*sin(x).