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A198755 Decimal expansion of x>0 satisfying x^2+cos(x)=2. 106
1, 3, 2, 5, 6, 2, 2, 5, 1, 8, 1, 4, 7, 5, 3, 6, 6, 2, 3, 4, 8, 3, 2, 2, 9, 0, 2, 9, 3, 8, 7, 9, 8, 7, 4, 4, 3, 3, 0, 4, 5, 4, 6, 7, 2, 5, 6, 5, 7, 6, 6, 4, 9, 5, 2, 6, 2, 7, 4, 0, 1, 8, 5, 3, 2, 0, 0, 8, 9, 5, 0, 6, 1, 6, 5, 9, 3, 0, 2, 4, 6, 5, 0, 3, 4, 1, 1, 0, 9, 7, 5, 9, 7, 7, 5, 7, 5, 6, 7 (list; constant; graph; refs; listen; history; text; internal format)
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.
LINKS
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
Sequence in context: A343342 A338470 A262395 * A134237 A341521 A227192
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Oct 30 2011
STATUS
approved

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Last modified April 25 11:03 EDT 2024. Contains 371967 sequences. (Running on oeis4.)