A199072 -id:A199072 - cp's OEIS frontend

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

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

A199071 Decimal expansion of x<0 satisfying 2x^2+3sin(x)=1.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Nov 02 2011

See A198866 for a guide to related sequences. The Mathematica program includes a graph.

			negative: -1.40710663976303990599616688903595712850...
positive:  0.28352555144412274861845117327862135032...

Index entries for transcendental numbers.

Cf. A198866.

a = 2; b = 3; 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, -1.5, -1.4}, WorkingPrecision -> 110]
RealDigits[r] (* A199071 *)
r = x /. FindRoot[f[x] == g[x], {x, .28, .29}, WorkingPrecision -> 110]
RealDigits[r] (* A199072 *)

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

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A199071 Decimal expansion of x<0 satisfying 2x^2+3sin(x)=1.

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cp's OEIS Frontend

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

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

A199071 Decimal expansion of x<0 satisfying 2*x^2+3*sin(x)=1.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A199071 Decimal expansion of x<0 satisfying 2x^2+3sin(x)=1.