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%I #12 Mar 30 2012 18:57:56
%S 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,
%T 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,
%U 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
%N Decimal expansion of x>0 satisfying x^2+cos(x)=2.
%C For many choices of a,b,c, there is a unique x>0 satisfying a*x^2+b*cos(x)=c.
%C Guide to related sequences, with graphs included in Mathematica programs:
%C a.... b.... c..... x
%C 1.... 1.... 2..... A198755
%C 1.... 1.... 3..... A198756
%C 1.... 1.... 4..... A198757
%C 1.... 2.... 3..... A198758
%C 1.... 2.... 4..... A198811
%C 1.... 3.... 3..... A198812
%C 1.... 3.... 4..... A198813
%C 1.... 4.... 3..... A198814
%C 1.... 4.... 4..... A198815
%C 1.... 1.... 0..... A125578
%C 1... -1.... 1..... A198816
%C 1... -1.... 2..... A198817
%C 1... -1.... 3..... A198818
%C 1... -1.... 4..... A198819
%C 1... -2.... 1..... A198821
%C 1... -2.... 2..... A198822
%C 1... -2.... 3..... A198823
%C 1... -2.... 4..... A198824
%C 1... -2... -1..... A198825
%C 1... -3.... 0..... A197807
%C 1... -3.... 1..... A198826
%C 1... -3.... 2..... A198828
%C 1... -3.... 3..... A198829
%C 1... -3.... 4..... A198830
%C 1... -3... -1..... A198835
%C 1... -3... -2..... A198836
%C 1... -4.... 0..... A197808
%C 1... -4.... 1..... A198838
%C 1... -4.... 2..... A198839
%C 1... -4.... 3..... A198840
%C 1... -4.... 4..... A198841
%C 1... -4... -1..... A198842
%C 1... -4... -2..... A198843
%C 1... -4... -3..... A198844
%C 2.... 0.... 1..... A010503
%C 2.... 0.... 3..... A115754
%C 2.... 1.... 2..... A198820
%C 2.... 1.... 3..... A198827
%C 2.... 1.... 4..... A198837
%C 2.... 2.... 3..... A198869
%C 2.... 3.... 4..... A198870
%C 2... -1.... 1..... A198871
%C 2... -1.... 2..... A198872
%C 2... -1.... 3..... A198873
%C 2... -1.... 4..... A198874
%C 2... -2... -1..... A198875
%C 2... -2.... 3..... A198876
%C 2... -3... -2..... A198877
%C 2... -3... -1..... A198878
%C 2... -3.... 1..... A198879
%C 2... -3.... 2..... A198880
%C 2... -3.... 3..... A198881
%C 2... -3.... 4..... A198882
%C 2... -4... -3..... A198883
%C 2... -4... -1..... A198884
%C 2... -4.... 1..... A198885
%C 2... -4.... 3..... A198886
%C 3.... 0.... 1..... A020760
%C 3.... 1.... 2..... A198868
%C 3.... 1.... 3..... A198917
%C 3.... 1.... 4..... A198918
%C 3.... 2.... 3..... A198919
%C 3.... 2.... 4..... A198920
%C 3.... 3.... 4..... A198921
%C 3... -1.... 1..... A198922
%C 3... -1.... 2..... A198924
%C 3... -1.... 3..... A198925
%C 3... -1.... 4..... A198926
%C 3... -2... -1..... A198927
%C 3... -2.... 1..... A198928
%C 3... -2.... 2..... A198929
%C 3... -2.... 3..... A198930
%C 3... -2.... 4..... A198931
%C 3... -3... -1..... A198932
%C 3... -3.... 1..... A198933
%C 3... -3.... 2..... A198934
%C 3... -3.... 4..... A198935
%C 3... -4... -3..... A198936
%C 3... -4... -2..... A198937
%C 3... -4... -1..... A198938
%C 3... -4.... 1..... A198939
%C 3... -4.... 2..... A198940
%C 3... -4.... 3..... A198941
%C 3... -4.... 4..... A198942
%C 4.... 1.... 2..... A198923
%C 4.... 1.... 3..... A198983
%C 4.... 1.... 4..... A198984
%C 4.... 2.... 3..... A198985
%C 4.... 3.... 4..... A198986
%C 4... -1.... 1..... A198987
%C 4... -1.... 2..... A198988
%C 4... -1.... 3..... A198989
%C 4... -1.... 4..... A198990
%C 4... -2... -1..... A198991
%C 4... -2.... 1..... A198992
%C 4... -2... -3..... A198993
%C 4... -3... -2..... A198994
%C 4... -3... -1..... A198995
%C 4... -2.... 1..... A198996
%C 4... -3.... 2..... A198997
%C 4... -3.... 3..... A198998
%C 4... -3.... 4..... A198999
%C 4... -4... -3..... A199000
%C 4... -4... -1..... A199001
%C 4... -4.... 1..... A199002
%C 4... -4.... 3..... A199003
%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 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.
%e x=1.32562251814753662348322902938798744330...
%t (* Program 1: A198655 *)
%t a = 1; b = 1; c = 2;
%t f[x_] := a*x^2 + b*Cos[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, 1.32, 1.33}, WorkingPrecision -> 110]
%t RealDigits[r] (* A198755 *)
%t (* Program 2: implicit surface of x^2+u*cos(x)=v *)
%t f[{x_, u_, v_}] := x^2 + u*Cos[x] - v;
%t t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 3}]}, {u, -5, 4}, {v, u, 20}];
%t ListPlot3D[Flatten[t, 1]] (* for A198755 *)
%Y Cf. A197737, A198414.
%K nonn,cons
%O 1,2
%A _Clark Kimberling_, Oct 30 2011
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