cp's OEIS Frontend

A199429 Decimal expansion of x>0 satisfying x^2+x*sin(x)=cos(x).

%I A199429 #15 Feb 07 2025 20:07:34
%S A199429 6,4,3,4,3,6,3,6,4,1,3,8,0,2,6,1,5,8,6,4,2,0,9,8,9,1,4,3,0,4,0,1,3,1,
%T A199429 8,2,6,8,7,4,4,6,7,2,4,1,9,4,5,7,8,5,1,6,3,2,3,8,7,4,9,1,9,8,5,8,8,7,
%U A199429 5,2,2,9,2,2,2,7,2,5,9,4,1,7,6,4,1,7,8,8,8,7,0,7,8,5,2,7,8,5,7
%N A199429 Decimal expansion of x>0 satisfying x^2+x*sin(x)=cos(x).
%C A199429 For many choices of a,b,c, there is exactly one x>0 satisfying a*x^2+b*x*sin(x)=c*cos(x).
%C A199429 Guide to related sequences, with graphs included in Mathematica programs:
%C A199429 a.... b.... c.... x
%C A199429 1.... 1.... 1.... A199429
%C A199429 1.... 1.... 2.... A199430
%C A199429 1.... 1.... 3.... A199431
%C A199429 1.... 2.... 1.... A199432
%C A199429 1.... 2.... 2.... A199433
%C A199429 1.... 2.... 3.... A199434
%C A199429 1.... 3.... 1.... A199435
%C A199429 1.... 3.... 2.... A199436
%C A199429 1.... 3.... 3.... A199437
%C A199429 2.... 1.... 1.... A199438
%C A199429 2.... 1.... 2.... A199439
%C A199429 2.... 1.... 3.... A199440
%C A199429 2.... 2.... 1.... A199441
%C A199429 2.... 2.... 3.... A199442
%C A199429 2.... 3.... 1.... A199443
%C A199429 2.... 3.... 2.... A199444
%C A199429 2.... 3.... 3.... A199445
%C A199429 2.... 1.... 1.... A199446
%C A199429 3.... 1.... 2.... A199447
%C A199429 3.... 1.... 3.... A199448
%C A199429 3.... 2.... 1.... A199449
%C A199429 3.... 2.... 2.... A199450
%C A199429 3.... 2.... 3.... A199451
%C A199429 3.... 3.... 1.... A199452
%C A199429 3.... 3.... 2.... A199453
%C A199429 1... -1.... 1.... A199454
%C A199429 1... -1.... 2.... A199455
%C A199429 1... -1.... 3.... A199456
%C A199429 1... -2... -3.... A199457
%C A199429 1... -2... -2.... A199458
%C A199429 1... -2... -1.... A199459
%C A199429 1... -2...  0.... A199460
%C A199429 1... -2...  1.... A199461
%C A199429 1... -2...  2.... A199462
%C A199429 1... -2...  3.... A199463
%C A199429 1... -3... -3.... A199464
%C A199429 1... -3... -2.... A199465
%C A199429 1... -3... -1.... A199466
%C A199429 1... -3...  0.... A199467
%C A199429 1... -3...  1.... A199468
%C A199429 1... -3...  2.... A199469
%C A199429 1... -3...  3.... A199470
%C A199429 2... -1...  1.... A199471
%C A199429 2... -1...  2.... A199472
%C A199429 2... -1...  3.... A199473
%C A199429 2... -2...  1.... A199503
%C A199429 2... -2...  3.... A199504
%C A199429 3... -1...  1.... A199505
%C A199429 2... -1...  2.... A199506
%C A199429 2... -1...  3.... A199507
%C A199429 2... -2...  1.... A199508
%C A199429 2... -2...  2.... A199509
%C A199429 2... -2...  3.... A199510
%C A199429 3... -3...  1.... A199511
%C A199429 3... -3...  2.... A199513
%C A199429 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 A199429 For an example related to A199429, take f(x,u,v)=x^2+u*x*sin(x)-v*cos(x) 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 A199429 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A199429 0.6434363641380261586420989143040131826874...
%t A199429 (* Program 1:  A199429 *)
%t A199429 a = 1; b = 1; c = 1;
%t A199429 f[x_] := a*x^2 + b*x*Sin[x]; g[x_] := c*Cos[x]
%t A199429 Plot[{f[x], g[x]}, {x, -2 Pi, 2 Pi}, {AxesOrigin -> {0, 0}}]
%t A199429 r = x /. FindRoot[f[x] == g[x], {x, .64, .65}, WorkingPrecision -> 110]
%t A199429 RealDigits[r]  (* A199429 *)
%t A199429 (* Program 2: implicit surface: x^2+u*x*sin(x)=v*cos(x) *)
%t A199429 f[{x_, u_, v_}] := x^2 + u*x*Sin[x] - v*Cos[x];
%t A199429 t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 10}, {v, u, 100}];
%t A199429 ListPlot3D[Flatten[t, 1]]  (* for A199429 *)
%o A199429 (PARI) g(a,b,c)=solve(x=0,abs(a)+abs(b)+abs(c), my(S=sin(x),C=sqrt(1-s^2)); a*x^2+b*x*S-c*C)
%o A199429 g(1,1,1) \\ _Charles R Greathouse IV_, Feb 07 2025
%Y A199429 Cf. A199370, A199170, A198866, A198755, A198414, A197737.
%K A199429 nonn,cons
%O A199429 0,1
%A A199429 _Clark Kimberling_, Nov 06 2011