%I #20 Jan 30 2025 11:47:19

%S 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,

%T 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,

%U 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

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

%C For many choices of a,b,c, there are exactly two numbers x having a*x^2 + b*sin(x) = c.

%C Guide to related sequences, with graphs included in Mathematica programs:

%C a.... b.... c.... x

%C 1.... 1.... 1.... A124597

%C 1.... 1.... 1.... A198866, A198867

%C 1.... 1.... 2.... A199046, A199047

%C 1.... 1.... 3.... A199048, A199049

%C 1.... 2.... 0.... A198414

%C 1.... 2.... 1.... A199080, A199081

%C 1.... 2.... 2.... A199082, A199083

%C 1.... 2.... 3.... A199050, A199051

%C 1.... 3.... 0.... A198415

%C 1.... 3... -1.... A199052, A199053

%C 1.... 3.... 1.... A199054, A199055

%C 1.... 3.... 2.... A199056, A199057

%C 1.... 3.... 3.... A199058, A199059

%C 2.... 1.... 0.... A198583

%C 2.... 1.... 1.... A199061, A199062

%C 2.... 1.... 2.... A199063, A199064

%C 2.... 1.... 3.... A199065, A199066

%C 2.... 2.... 1.... A199067, A199068

%C 2.... 2.... 3.... A199069, A199070

%C 2.... 3.... 0.... A198605

%C 2.... 3.... 1.... A199071, A199072

%C 2.... 3.... 2.... A199073, A199074

%C 2.... 3.... 3.... A199075, A199076

%C 3.... 0.... 1.... A020760

%C 3.... 1.... 1.... A199060, A199077

%C 3.... 1.... 2.... A199078, A199079

%C 3.... 1.... 3.... A199150, A199151

%C 3.... 2.... 1.... A199152, A199153

%C 3.... 2.... 2.... A199154, A199155

%C 3.... 2.... 3.... A199156, A199157

%C 3.... 3.... 1.... A199158, A199159

%C 3.... 3.... 2.... A199160, A199161

%C 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.

%C 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.

%H G. C. Greubel, <a href="/A198866/b198866.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%e negative: -1.40962400400259624923559397058949354...

%e positive: 0.63673265080528201088799090383828005...

%t (* Program 1: this sequence and A198867 *)

%t a = 1; b = 1; c = 1;

%t f[x_] := a*x^2 + b*Sin[x]; g[x_] := c

%t Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]

%t r = x /. FindRoot[f[x] == g[x], {x, -1.41, -1.40}, WorkingPrecision -> 110]

%t RealDigits[r] (* this sequence *)

%t r = x /. FindRoot[f[x] == g[x], {x, .63, .64}, WorkingPrecision -> 110]

%t RealDigits[r] (* A198867 *)

%t (* Program 2: implicit surface of x^2+u*sin(x)=v *)

%t f[{x_, u_, v_}] := x^2 + u*Sin[x] - v;

%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 6}, {v, u, 12}];

%t ListPlot3D[Flatten[t, 1]] (* for this sequence *)

%o (PARI) a=1; b=1; c=1; solve(x=-2, 0, a*x^2 + b*sin(x) - c) \\ _G. C. Greubel_, Feb 20 2019

%o (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

%Y Cf. A198867, A198755, A198414, A197737.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Nov 02 2011