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

Author

Clark Kimberling, Nov 02 2011

Keywords

• nonn
• cons

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.

Examples

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

Links

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

Crossrefs

Cf. A198867, A198755, A198414, A197737.

Programs

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 *)

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