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

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

1,2

Comments

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.... 1.... A124597

1.... 1.... 1.... A198866, A198867

1.... 1.... 2.... A199046, A199047

1.... 1.... 3.... A199048, A199049

1.... 2.... 0.... A198414

1.... 2.... 1.... A199080, A199081

1.... 2.... 2.... A199082, A199083

1.... 2.... 3.... A199050, A199051

1.... 3.... 0.... A198415

1.... 3... -1.... A199052, A199053

1.... 3.... 1.... A199054, A199055

1.... 3.... 2.... A199056, A199057

1.... 3.... 3.... A199058, A199059

2.... 1.... 0.... A198583

2.... 1.... 1.... A199061, A199062

2.... 1.... 2.... A199063, A199064

2.... 1.... 3.... A199065, A199066

2.... 2.... 1.... A199067, A199068

2.... 2.... 3.... A199069, A199070

2.... 3.... 0.... A198605

2.... 3.... 1.... A199071, A199072

2.... 3.... 2.... A199073, A199074

2.... 3.... 3.... A199075, A199076

3.... 0.... 1.... A020760

3.... 1.... 1.... A199060, A199077

3.... 1.... 2.... A199078, A199079

3.... 1.... 3.... A199150, A199151

3.... 2.... 1.... A199152, A199153

3.... 2.... 2.... A199154, A199155

3.... 2.... 3.... A199156, A199157

3.... 3.... 1.... A199158, A199159

3.... 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.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Example

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

Mathematica

(* 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 *)

Prog

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

Crossrefs

Cf. A198867, A198755, A198414, A197737.

Sequence in context: A215499 A190262 A187586 * A269720 A245638 A176220

Adjacent sequences: A198863 A198864 A198865 * A198867 A198868 A198869

Keyword

nonn,cons

Author

Clark Kimberling, Nov 02 2011

Status

approved