A198414 -id:A198414 - 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

A198755 Decimal expansion of x>0 satisfying x^2+cos(x)=2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 30 2011

For many choices of a,b,c, there is a unique x>0 satisfying a*x^2+b*cos(x)=c.
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c..... x
... 1.... 2..... A198755
... 1.... 3..... A198756
... 1.... 4..... A198757
... 2.... 3..... A198758
... 2.... 4..... A198811
... 3.... 3..... A198812
... 3.... 4..... A198813
... 4.... 3..... A198814
... 4.... 4..... A198815
... 1.... 0..... A125578
.. -1.... 1..... A198816
.. -1.... 2..... A198817
.. -1.... 3..... A198818
.. -1.... 4..... A198819
.. -2.... 1..... A198821
.. -2.... 2..... A198822
.. -2.... 3..... A198823
.. -2.... 4..... A198824
.. -2... -1..... A198825
.. -3.... 0..... A197807
.. -3.... 1..... A198826
.. -3.... 2..... A198828
.. -3.... 3..... A198829
.. -3.... 4..... A198830
.. -3... -1..... A198835
.. -3... -2..... A198836
.. -4.... 0..... A197808
.. -4.... 1..... A198838
.. -4.... 2..... A198839
.. -4.... 3..... A198840
.. -4.... 4..... A198841
.. -4... -1..... A198842
.. -4... -2..... A198843
.. -4... -3..... A198844
... 0.... 1..... A010503
... 0.... 3..... A115754
... 1.... 2..... A198820
... 1.... 3..... A198827
... 1.... 4..... A198837
... 2.... 3..... A198869
... 3.... 4..... A198870
.. -1.... 1..... A198871
.. -1.... 2..... A198872
.. -1.... 3..... A198873
.. -1.... 4..... A198874
.. -2... -1..... A198875
.. -2.... 3..... A198876
.. -3... -2..... A198877
.. -3... -1..... A198878
.. -3.... 1..... A198879
.. -3.... 2..... A198880
.. -3.... 3..... A198881
.. -3.... 4..... A198882
.. -4... -3..... A198883
.. -4... -1..... A198884
.. -4.... 1..... A198885
.. -4.... 3..... A198886
... 0.... 1..... A020760
... 1.... 2..... A198868
... 1.... 3..... A198917
... 1.... 4..... A198918
... 2.... 3..... A198919
... 2.... 4..... A198920
... 3.... 4..... A198921
.. -1.... 1..... A198922
.. -1.... 2..... A198924
.. -1.... 3..... A198925
.. -1.... 4..... A198926
.. -2... -1..... A198927
.. -2.... 1..... A198928
.. -2.... 2..... A198929
.. -2.... 3..... A198930
.. -2.... 4..... A198931
.. -3... -1..... A198932
.. -3.... 1..... A198933
.. -3.... 2..... A198934
.. -3.... 4..... A198935
.. -4... -3..... A198936
.. -4... -2..... A198937
.. -4... -1..... A198938
.. -4.... 1..... A198939
.. -4.... 2..... A198940
.. -4.... 3..... A198941
.. -4.... 4..... A198942
... 1.... 2..... A198923
... 1.... 3..... A198983
... 1.... 4..... A198984
... 2.... 3..... A198985
... 3.... 4..... A198986
.. -1.... 1..... A198987
.. -1.... 2..... A198988
.. -1.... 3..... A198989
.. -1.... 4..... A198990
.. -2... -1..... A198991
.. -2.... 1..... A198992
.. -2... -3..... A198993
.. -3... -2..... A198994
.. -3... -1..... A198995
.. -2.... 1..... A198996
.. -3.... 2..... A198997
.. -3.... 3..... A198998
.. -3.... 4..... A198999
.. -4... -3..... A199000
.. -4... -1..... A199001
.. -4.... 1..... A199002
.. -4.... 3..... A199003
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 A198755, take f(x,u,v)=x^2+u*cos(x)-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.

			1.32562251814753662348322902938798744330...

Index entries for transcendental numbers.

Cf. A197737, A198414.

(* Program 1:  A198655 *)
a = 1; b = 1; c = 2;
f[x_] := a*x^2 + b*Cos[x]; g[x_] := c
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.32, 1.33}, WorkingPrecision -> 110]
RealDigits[r] (* A198755 *)
(* Program 2: implicit surface of x^2+u*cos(x)=v *)
f[{x_, u_, v_}] := x^2 + u*Cos[x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 3}]}, {u, -5, 4}, {v, u, 20}];
ListPlot3D[Flatten[t, 1]]  (* for A198755 *)

A199597 Decimal expansion of x > 0 satisfying x^2 + x*cos(x) = sin(x).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Nov 08 2011

For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*cos(x)=c*sin(x).
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
... 1.... 2.... A199597
... 1.... 3.... A199598
... 1.... 4.... A199599
... 2.... 1.... A199600
... 2.... 3.... A199601
... 2.... 4.... A199602
... 3.... 0.... A199603, A199604
... 3.... 1.... A199605, A199606
... 3.... 2.... A199607, A199608
... 3.... 3.... A199609, A199610
... 4.... 0.... A199611, A199612
... 4.... 1.... A199613, A199614
... 4.... 2.... A199615, A199616
... 4.... 3.... A199617, A199618
... 4.... 4.... A199619, A199620
... 1.... 0.... A199621
... 1.... 2.... A199622
... 1.... 3.... A199623
... 1.... 4.... A199624
... 2.... 1.... A199625
... 2.... 3.... A199661
... 1.... 0.... A199662
... 1.... 2.... A199663
... 1.... 3.... A199664
... 1.... 4.... A199665
... 2.... 0.... A199666
... 2.... 1.... A199667
... 2.... 3.... A199668
... 2.... 4.... A199669
.. -1.... 0.... A003957
.. -1.... 1.... A199722
.. -1.... 2.... A199721
.. -1.... 3.... A199720
.. -1.... 4.... A199719
.. -2.... 1.... A199726
.. -2.... 2.... A199725
.. -2.... 3.... A199724
.. -2.... 4.... A199723
.. -3.... 1.... A199730
.. -3.... 2.... A199729
.. -3.... 3.... A199728
.. -3.... 4.... A199727
.. -4.... 1.... A199737. A199738
.. -4.... 2.... A199735, A199736
.. -4.... 3.... A199733, A199734
.. -4.... 4.... A199731. A199732
.. -1.... 1.... A199742
.. -1.... 2.... A199741
.. -1.... 3.... A199740
.. -1.... 4.... A199739
.. -2.... 1.... A199776
.. -2.... 3.... A199775
.. -3.... 1.... A199780
.. -3.... 2.... A199779
.. -3.... 3.... A199778
.. -3.... 4.... A199777
.. -4.... 1.... A199782
.. -4.... 3.... A199781
.. -4.... 1.... A199786
.. -4.... 2.... A199785
.. -4.... 3.... A199784
.. -4.... 4.... A199783
.. -3.... 1.... A199789
.. -3.... 2.... A199788
.. -3.... 4.... A199787
.. -2.... 1.... A199793
.. -2.... 2.... A199792
.. -2.... 3.... A199791
.. -2.... 4.... A199790
.. -1.... 1.... A199797
.. -1.... 2.... A199796
.. -1.... 3.... A199795
.. -1.... 4.... A199794
.. -4.... 1.... A199873
.. -4.... 3.... A199872
.. -3.... 1.... A199871
.. -3.... 2.... A199870
.. -3.... 3.... A199869
.. -3.... 4.... A199868
.. -2.... 1.... A199867
.. -2.... 3.... A199866
.. -1.... 1.... A199865
.. -1.... 2.... A199864
.. -1.... 3.... A199863
.. -1.... 4.... A199862
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^2+u*x*cos(x)-v*sin(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.

			1.1881851344514388032178109729076525973...

Index entries for transcendental numbers.

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

(* Program 1:  A199597 *)
a = 1; b = 1; c = 2;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -Pi, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.18, 1.19}, WorkingPrecision -> 110]
RealDigits[r]  (* A199597 *)
(* Program 2: impl. surf. x^2+u*x*cos(x)=v*sin(x) *)
f[{x_, u_, v_}] := x^2 + u*x*Cos[x] - v*Sin[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, .5, 3}]}, {u, 0, 2}, {v, u, 20}];
ListPlot3D[Flatten[t, 1]]  (* for A199597 *)

Edited by Georg Fischer, Aug 03 2021

A198866 Decimal expansion of x < 0 satisfying x^2 + sin(x) = 1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Nov 02 2011

For many choices of a,b,c, there are exactly two numbers x having a*x^2 + b*sin(x) = c.
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
... 1.... 1.... A124597
... 1.... 1.... A198866, A198867
... 1.... 2.... A199046, A199047
... 1.... 3.... A199048, A199049
... 2.... 0.... A198414
... 2.... 1.... A199080, A199081
... 2.... 2.... A199082, A199083
... 2.... 3.... A199050, A199051
... 3.... 0.... A198415
... 3... -1.... A199052, A199053
... 3.... 1.... A199054, A199055
... 3.... 2.... A199056, A199057
... 3.... 3.... A199058, A199059
... 1.... 0.... A198583
... 1.... 1.... A199061, A199062
... 1.... 2.... A199063, A199064
... 1.... 3.... A199065, A199066
... 2.... 1.... A199067, A199068
... 2.... 3.... A199069, A199070
... 3.... 0.... A198605
... 3.... 1.... A199071, A199072
... 3.... 2.... A199073, A199074
... 3.... 3.... A199075, A199076
... 0.... 1.... A020760
... 1.... 1.... A199060, A199077
... 1.... 2.... A199078, A199079
... 1.... 3.... A199150, A199151
... 2.... 1.... A199152, A199153
... 2.... 2.... A199154, A199155
... 2.... 3.... A199156, A199157
... 3.... 1.... A199158, A199159
... 3.... 2.... A199160, A199161
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 A198866, take f(x,u,v) = x^2 + u*sin(x) - 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.

			negative: -1.40962400400259624923559397058949354...
positive:  0.63673265080528201088799090383828005...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers.

Cf. A198867, A198755, A198414, A197737.

(* Program 1: this sequence and A198867 *)
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*Sin[x]; g[x_] := c
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1.41, -1.40}, WorkingPrecision -> 110]
RealDigits[r] (* this sequence *)
r = x /. FindRoot[f[x] == g[x], {x, .63, .64}, WorkingPrecision -> 110]
RealDigits[r] (* A198867 *)
(* Program 2: implicit surface of x^2+u*sin(x)=v *)
f[{x_, u_, v_}] := x^2 + u*Sin[x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 6}, {v, u, 12}];
ListPlot3D[Flatten[t, 1]]  (* for this sequence *)

a=1; b=1; c=1; solve(x=-2, 0, a*x^2 + b*sin(x) - c) \\ G. C. Greubel, Feb 20 2019

a=1; b=1; c=1; (a*x^2 + b*sin(x)==c).find_root(-2,0,x) # G. C. Greubel, Feb 20 2019

A199429 Decimal expansion of x>0 satisfying x^2+x*sin(x)=cos(x).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Nov 06 2011

For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*sin(x)=c*cos(x).
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
... 1.... 1.... A199429
... 1.... 2.... A199430
... 1.... 3.... A199431
... 2.... 1.... A199432
... 2.... 2.... A199433
... 2.... 3.... A199434
... 3.... 1.... A199435
... 3.... 2.... A199436
... 3.... 3.... A199437
... 1.... 1.... A199438
... 1.... 2.... A199439
... 1.... 3.... A199440
... 2.... 1.... A199441
... 2.... 3.... A199442
... 3.... 1.... A199443
... 3.... 2.... A199444
... 3.... 3.... A199445
... 1.... 1.... A199446
... 1.... 2.... A199447
... 1.... 3.... A199448
... 2.... 1.... A199449
... 2.... 2.... A199450
... 2.... 3.... A199451
... 3.... 1.... A199452
... 3.... 2.... A199453
.. -1.... 1.... A199454
.. -1.... 2.... A199455
.. -1.... 3.... A199456
.. -2... -3.... A199457
.. -2... -2.... A199458
.. -2... -1.... A199459
.. -2...  0.... A199460
.. -2...  1.... A199461
.. -2...  2.... A199462
.. -2...  3.... A199463
.. -3... -3.... A199464
.. -3... -2.... A199465
.. -3... -1.... A199466
.. -3...  0.... A199467
.. -3...  1.... A199468
.. -3...  2.... A199469
.. -3...  3.... A199470
.. -1...  1.... A199471
.. -1...  2.... A199472
.. -1...  3.... A199473
.. -2...  1.... A199503
.. -2...  3.... A199504
.. -1...  1.... A199505
.. -1...  2.... A199506
.. -1...  3.... A199507
.. -2...  1.... A199508
.. -2...  2.... A199509
.. -2...  3.... A199510
.. -3...  1.... A199511
.. -3...  2.... A199513
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 A199429, take f(x,u,v)=x^2+u*x*sin(x)-v*cos(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.

			0.6434363641380261586420989143040131826874...

Index entries for transcendental numbers.

Cf. A199370, A199170, A198866, A198755, A198414, A197737.

(* Program 1:  A199429 *)
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*x*Sin[x]; g[x_] := c*Cos[x]
Plot[{f[x], g[x]}, {x, -2 Pi, 2 Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .64, .65}, WorkingPrecision -> 110]
RealDigits[r]  (* A199429 *)
(* Program 2: implicit surface: x^2+u*x*sin(x)=v*cos(x) *)
f[{x_, u_, v_}] := x^2 + u*x*Sin[x] - v*Cos[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 10}, {v, u, 100}];
ListPlot3D[Flatten[t, 1]]  (* for A199429 *)

g(a,b,c)=solve(x=0,abs(a)+abs(b)+abs(c), my(S=sin(x),C=sqrt(1-s^2)); a*x^2+b*x*S-c*C)
g(1,1,1) \\ Charles R Greathouse IV, Feb 07 2025

A199170 Decimal expansion of x<0 satisfying x^2+x*cos(x)=1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Nov 03 2011

For many choices of a,b,c, there are exactly two numbers x satisfying a*x^2+b*x*cos(x)=c.
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
... 1.... 1.... A199170, A199171
... 1.... 2.... A199172, A199173
... 1.... 3.... A199174, A199175
... 2.... 1.... A199176, A199177
... 2.... 2.... A199178, A199179
... 2.... 3.... A199180, A199181
... 3.... 1.... A199182, A199183
... 3.... 2.... A199184, A199185
... 3.... 3.... A199186, A199187
... 1.... 1.... A199188, A199189
... 1.... 2.... A199265, A199266
... 1.... 3.... A199267, A199268
... 2.... 1.... A199269, A199270
... 2.... 3.... A199271, A199272
... 3.... 1.... A199273, A199274
... 3.... 2.... A199275, A199276
... 3.... 3.... A199277, A199278
... 1.... 1.... A199279, A199280
... 1.... 2.... A199281, A199282
... 1.... 3.... A199283, A199284
... 2.... 1.... A199285, A199286
... 2.... 2.... A199287, A199288
... 2.... 3.... A199289, A199290
... 3.... 1.... A199291, A199292
... 3.... 2.... A199293, A199294
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 A199170, take f(x,u,v)=x^2+u*xcos(x)-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.

			negative: -1.19835984451866026826502160343030898927268...
positive:  0.685174133854503187895211530638458709591...

Index entries for transcendental numbers.

Cf. A199171, A197737, A198414, A198755, A198866.

(* Program 1:  A199170 and A199171 *)
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*x*Cos[x]; g[x_] := c
Plot[{f[x], g[x]}, {x, -2 Pi, 2 Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1.2, -1.1}, WorkingPrecision -> 110]
RealDigits[r]  (* A199170 *)
r = x /. FindRoot[f[x] == g[x], {x, .68, .69}, WorkingPrecision -> 110]
RealDigits[r]  (* A199171 *)
(* Program 2: implicit surface of x^2+u*x*cos(x)=v *)
f[{x_, u_, v_}] := x^2 + u*x*Cos[x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0,
    1.9}, {v, u, 600}];
ListPlot3D[Flatten[t, 1]]  (* for A199170 *)

A199370 Decimal expansion of x>0 satisfying x^2+x*sin(x)=1.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Nov 05 2011

For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*sin(x)=c.
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
... 1.... 1.... A199370
... 1.... 2.... A199371
... 1.... 3.... A199372
... 2.... 1.... A199373
... 2.... 2.... A199374
... 2.... 3.... A199375
... 3.... 1.... A199376
... 3.... 2.... A199377
... 3.... 3.... A199378
... 1.... 1.... A199379
... 1.... 2.... A199180
... 1.... 3.... A199181
... 2.... 1.... A199182
... 2.... 3.... A199183
... 3.... 1.... A199184
... 3.... 2.... A199185
... 3.... 3.... A199186
... 1.... 1.... A199187
... 1.... 2.... A199188
... 1.... 3.... A199189
... 2.... 1....
... 2.... 2....
... 2.... 3....
... 3.... 1....
... 3.... 2....
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 A199370, take f(x,u,v)=x^2+u*x*sin(x)-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.

			0.722587549922524768355932872877196755159...

Index entries for transcendental numbers.

Cf. A199371, A199170, A198866, A198755, A198414, A197737.

(* Program 1: A199370 *)
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*x*Sin[x]; g[x_] := c
Plot[{f[x], g[x]}, {x, -1, Pi}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .72, .73}, WorkingPrecision -> 110]
RealDigits[r]  (* A199370 *)
(* Program 2: implicit surface of x^2+u*x*sin(x)=v *)
f[{x_, u_, v_}] := x^2 + u*x*Sin[x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 2.9}, {v, u, 600}];
ListPlot3D[Flatten[t, 1]]  (* for A199370 *)

A198415 Decimal expansion of x>0 having x^2 = 3*sin(x).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 24 2011

See A198414 for a guide to related sequences. The Mathematica program includes a graph.

			x=1.72212511207672359941514007347177405819...

Index entries for transcendental numbers.

Cf. A198414.

Digits:= 140:
fsolve(x^2-3*sin(x), x=1..2);  # Alois P. Heinz, Jan 29 2025

a = 1; b = 0; c = 3;
f[x_] := a*x^2 + b*x; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -1, 2}]
r = x /. FindRoot[f[x] == g[x], {x, 1.7, 1.8}, WorkingPrecision -> 110]
RealDigits[r] (* A198415 *)

solve(x=1,2, x^2-3*sin(x)) \\ Charles R Greathouse IV, Jan 28 2025

A198583 Decimal expansion of x>0 satisfying 2*x^2=sin(x).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Oct 27 2011

See A198414 for a guide to related sequences. The Mathematica program includes a graph.

			0.480945957728129657766297856537656383385407...

Index entries for transcendental numbers.

Cf. A198414.

a = 2; b = 0; c = 1;
f[x_] := a*x^2 + b*x; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -.1, 1}]
r = x /. FindRoot[f[x] == g[x], {x, .48, .49}, WorkingPrecision -> 110]
RealDigits[r] (* A198583 *)

A198605 Decimal expansion of x>0 satisfying 2x^2=3sin(x).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 27 2011

See A198414 for a guide to related sequences. The Mathematica program includes a graph.

			1.176921059464366442841871452208606113369...

Index entries for transcendental numbers.

Cf. A198414.

a = 2; b = 0; c = 3;
f[x_] := a*x^2 + b*x; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -.1, 1.3}]
r = x /. FindRoot[f[x] == g[x], {x, 1.1, 1.2}, WorkingPrecision -> 110]
RealDigits[r] (* A198605 *)

solve(x=1,2,3*sin(x)-2*x^2) \\ Charles R Greathouse IV, Apr 18 2016

cp's OEIS Frontend

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

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A198755 Decimal expansion of x>0 satisfying x^2+cos(x)=2.

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A199597 Decimal expansion of x > 0 satisfying x^2 + x*cos(x) = sin(x).

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A198866 Decimal expansion of x < 0 satisfying x^2 + sin(x) = 1.

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Sage

A199429 Decimal expansion of x>0 satisfying x^2+x*sin(x)=cos(x).

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A199170 Decimal expansion of x<0 satisfying x^2+x*cos(x)=1.

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A199370 Decimal expansion of x>0 satisfying x^2+x*sin(x)=1.

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A198415 Decimal expansion of x>0 having x^2 = 3*sin(x).

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A198605 Decimal expansion of x>0 satisfying 2x^2=3sin(x).