login
A198866
Decimal expansion of x < 0 satisfying x^2 + sin(x) = 1.
57
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
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
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Nov 02 2011
STATUS
approved