OFFSET
1,2
COMMENTS
For many choices of a,b,c, there is a unique x>0 satisfying a*x^2+b*cos(x)=c.
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c..... x
1.... 1.... 2..... A198755
1.... 1.... 3..... A198756
1.... 1.... 4..... A198757
1.... 2.... 3..... A198758
1.... 2.... 4..... A198811
1.... 3.... 3..... A198812
1.... 3.... 4..... A198813
1.... 4.... 3..... A198814
1.... 4.... 4..... A198815
1.... 1.... 0..... A125578
1... -1.... 1..... A198816
1... -1.... 2..... A198817
1... -1.... 3..... A198818
1... -1.... 4..... A198819
1... -2.... 1..... A198821
1... -2.... 2..... A198822
1... -2.... 3..... A198823
1... -2.... 4..... A198824
1... -2... -1..... A198825
1... -3.... 0..... A197807
1... -3.... 1..... A198826
1... -3.... 2..... A198828
1... -3.... 3..... A198829
1... -3.... 4..... A198830
1... -3... -1..... A198835
1... -3... -2..... A198836
1... -4.... 0..... A197808
1... -4.... 1..... A198838
1... -4.... 2..... A198839
1... -4.... 3..... A198840
1... -4.... 4..... A198841
1... -4... -1..... A198842
1... -4... -2..... A198843
1... -4... -3..... A198844
2.... 0.... 1..... A010503
2.... 0.... 3..... A115754
2.... 1.... 2..... A198820
2.... 1.... 3..... A198827
2.... 1.... 4..... A198837
2.... 2.... 3..... A198869
2.... 3.... 4..... A198870
2... -1.... 1..... A198871
2... -1.... 2..... A198872
2... -1.... 3..... A198873
2... -1.... 4..... A198874
2... -2... -1..... A198875
2... -2.... 3..... A198876
2... -3... -2..... A198877
2... -3... -1..... A198878
2... -3.... 1..... A198879
2... -3.... 2..... A198880
2... -3.... 3..... A198881
2... -3.... 4..... A198882
2... -4... -3..... A198883
2... -4... -1..... A198884
2... -4.... 1..... A198885
2... -4.... 3..... A198886
3.... 0.... 1..... A020760
3.... 1.... 2..... A198868
3.... 1.... 3..... A198917
3.... 1.... 4..... A198918
3.... 2.... 3..... A198919
3.... 2.... 4..... A198920
3.... 3.... 4..... A198921
3... -1.... 1..... A198922
3... -1.... 2..... A198924
3... -1.... 3..... A198925
3... -1.... 4..... A198926
3... -2... -1..... A198927
3... -2.... 1..... A198928
3... -2.... 2..... A198929
3... -2.... 3..... A198930
3... -2.... 4..... A198931
3... -3... -1..... A198932
3... -3.... 1..... A198933
3... -3.... 2..... A198934
3... -3.... 4..... A198935
3... -4... -3..... A198936
3... -4... -2..... A198937
3... -4... -1..... A198938
3... -4.... 1..... A198939
3... -4.... 2..... A198940
3... -4.... 3..... A198941
3... -4.... 4..... A198942
4.... 1.... 2..... A198923
4.... 1.... 3..... A198983
4.... 1.... 4..... A198984
4.... 2.... 3..... A198985
4.... 3.... 4..... A198986
4... -1.... 1..... A198987
4... -1.... 2..... A198988
4... -1.... 3..... A198989
4... -1.... 4..... A198990
4... -2... -1..... A198991
4... -2.... 1..... A198992
4... -2... -3..... A198993
4... -3... -2..... A198994
4... -3... -1..... A198995
4... -2.... 1..... A198996
4... -3.... 2..... A198997
4... -3.... 3..... A198998
4... -3.... 4..... A198999
4... -4... -3..... A199000
4... -4... -1..... A199001
4... -4.... 1..... A199002
4... -4.... 3..... A199003
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 A198755, take f(x,u,v)=x^2+u*cos(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.
EXAMPLE
x=1.32562251814753662348322902938798744330...
MATHEMATICA
(* Program 1: A198655 *)
a = 1; b = 1; c = 2;
f[x_] := a*x^2 + b*Cos[x]; g[x_] := c
Plot[{f[x], g[x]}, {x, -2, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.32, 1.33}, WorkingPrecision -> 110]
RealDigits[r] (* A198755 *)
(* Program 2: implicit surface of x^2+u*cos(x)=v *)
f[{x_, u_, v_}] := x^2 + u*Cos[x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 3}]}, {u, -5, 4}, {v, u, 20}];
ListPlot3D[Flatten[t, 1]] (* for A198755 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Oct 30 2011
STATUS
approved