A125578 -id:A125578 - cp's OEIS frontend

A197737 Decimal expansion of x<0 having x^2+x=cos(x).

1, 2, 5, 1, 1, 5, 1, 8, 3, 5, 2, 2, 0, 7, 6, 4, 8, 1, 1, 5, 9, 2, 8, 7, 0, 0, 6, 8, 7, 8, 8, 1, 6, 1, 8, 5, 9, 9, 4, 5, 3, 5, 6, 1, 0, 8, 5, 8, 8, 9, 6, 8, 6, 3, 6, 2, 0, 1, 7, 8, 2, 8, 1, 2, 1, 0, 3, 6, 0, 1, 9, 1, 8, 2, 3, 8, 2, 1, 0, 9, 1, 0, 4, 1, 1, 2, 7, 3, 5, 7, 6, 5, 9, 4, 8, 6, 8, 4, 2

Offset: 1

Clark Kimberling, Oct 20 2011

For many choices of a,b,c, there are exactly two numbers x having a*x^2+b*x=cos(x).
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
... 0.... 1.... A125578
... 0.... 2.... A197806
... 0.... 3.... A197807
... 0.... 4.... A197808
... 1.... 1.... A197737, A197738
... 1.... 2.... A197809, A197810
... 1.... 3.... A197811, A197812
... 1.... 4.... A197813, A197814
.. -2... -1.... A197815, A197820
.. -3... -1.... A197825, A197831
.. -4... -1.... A197839, A197840
... 2.... 1.... A197841, A197842
... 2.... 2.... A197843, A197844
... 2.... 3.... A197845, A197846
... 2.... 4.... A197847, A197848
.. -2... -2.... A197849, A197850
.. -3... -2.... A198098, A198099
.. -4... -2.... A198100, A198101
... 3.... 1.... A198102, A198103
... 3.... 2.... A198104, A198105
... 3.... 3.... A198106, A198107
... 3.... 4.... A198108, A198109
.. -2... -3.... A198140, A198141
.. -3... -3.... A198142, A198143
.. -4... -3.... A198144, A198145
... 0.... 1.... A198110
... 0.... 3.... A198111
... 1.... 1.... A198112, A198113
... 1.... 2.... A198114, A198115
... 1.... 3.... A198116, A198117
... 1.... 4.... A198118, A198119
... 1... -1.... A198120, A198121
.. -4... -1.... A198122, A198123
... 2.... 1.... A198124, A198125
... 2.... 3.... A198126, A198127
... 3.... 1.... A198128, A198129
... 3.... 2.... A198130, A198131
... 3.... 3.... A198132, A198133
... 3.... 4.... A198134, A198135
.. -4... -3.... A198136, A198137
... 0.... 1.... A198211
... 0.... 2.... A198212
... 0.... 4.... A198213
... 1.... 1.... A198214, A198215
... 1.... 2.... A198216, A198217
... 1.... 3.... A198218, A198219
... 1.... 4.... A198220, A198221
... 2.... 1.... A198222, A198223
... 2.... 2.... A198224, A198225
... 2.... 3.... A198226, A198227
... 2.... 4.... A198228, A198229
... 3.... 1.... A198230, A198231
... 3.... 2.... A198232, A198233
... 3.... 4.... A198234, A198235
... 4.... 1.... A198236, A198237
... 4.... 2.... A198238, A198239
... 4.... 3.... A198240, A198241
... 4.... 4.... A198138, A198139
.. -4... -1.... A198345, A198346
... 0.... 1.... A198347
... 0.... 3.... A198348
... 1.... 1.... A198349, A198350
... 1.... 2.... A198351, A198352
... 1.... 3.... A198353, A198354
... 1.... 4.... A198355, A198356
... 2.... 1.... A198357, A198358
... 2.... 3.... A198359, A198360
... 3.... 1.... A198361, A198362
... 3.... 2.... A198363, A198364
... 3.... 3.... A198365, A198366
... 3.... 4.... A198367, A198368
... 4.... 1.... A198369, A198370
... 4.... 3.... A198371, A198372
.. -4... -1.... A198373, A198374
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 A197737, take f(x,u,v)=x^2+u*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.

			negative: -1.25115183522076481159287006878816185994...
positive:  0.55000934992726156666495361947172926116...

Index entries for transcendental numbers.

Cf. A197738.

(* Program 1:  A197738 *)
a = 1; b = 1; c = 1;
f[x_] := a*x^2 + b*x; g[x_] := c*Cos[x]
Plot[{f[x], g[x]}, {x, -2, 1}]
r1 = x /. FindRoot[f[x] == g[x], {x, -1.26, -1.25}, WorkingPrecision -> 110]
RealDigits[r1] (* A197737 *)
r1 = x /. FindRoot[f[x] == g[x], {x, .55, .551}, WorkingPrecision -> 110]
RealDigits[r1] (* A197738 *)
(* Program 2: implicit surface of x^2+u*x=v*cos(x) *)
f[{x_, u_, v_}] := x^2 + u*x - v*Cos[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, 0, 20}, {v, u, 20}];
ListPlot3D[Flatten[t, 1]]  (* for A197737 *)

A197737_vec(N=150)={localprec(N+10); digits(solve(x=-1.5,-1,x^2+x-cos(x))\.1^N)} \\ M. F. Hasler, Aug 05 2021

A198755 Decimal expansion of x>0 satisfying x^2+cos(x)=2.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Oct 30 2011

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.... 2..... A198755
... 1.... 3..... A198756
... 1.... 4..... A198757
... 2.... 3..... A198758
... 2.... 4..... A198811
... 3.... 3..... A198812
... 3.... 4..... A198813
... 4.... 3..... A198814
... 4.... 4..... A198815
... 1.... 0..... A125578
.. -1.... 1..... A198816
.. -1.... 2..... A198817
.. -1.... 3..... A198818
.. -1.... 4..... A198819
.. -2.... 1..... A198821
.. -2.... 2..... A198822
.. -2.... 3..... A198823
.. -2.... 4..... A198824
.. -2... -1..... A198825
.. -3.... 0..... A197807
.. -3.... 1..... A198826
.. -3.... 2..... A198828
.. -3.... 3..... A198829
.. -3.... 4..... A198830
.. -3... -1..... A198835
.. -3... -2..... A198836
.. -4.... 0..... A197808
.. -4.... 1..... A198838
.. -4.... 2..... A198839
.. -4.... 3..... A198840
.. -4.... 4..... A198841
.. -4... -1..... A198842
.. -4... -2..... A198843
.. -4... -3..... A198844
... 0.... 1..... A010503
... 0.... 3..... A115754
... 1.... 2..... A198820
... 1.... 3..... A198827
... 1.... 4..... A198837
... 2.... 3..... A198869
... 3.... 4..... A198870
.. -1.... 1..... A198871
.. -1.... 2..... A198872
.. -1.... 3..... A198873
.. -1.... 4..... A198874
.. -2... -1..... A198875
.. -2.... 3..... A198876
.. -3... -2..... A198877
.. -3... -1..... A198878
.. -3.... 1..... A198879
.. -3.... 2..... A198880
.. -3.... 3..... A198881
.. -3.... 4..... A198882
.. -4... -3..... A198883
.. -4... -1..... A198884
.. -4.... 1..... A198885
.. -4.... 3..... A198886
... 0.... 1..... A020760
... 1.... 2..... A198868
... 1.... 3..... A198917
... 1.... 4..... A198918
... 2.... 3..... A198919
... 2.... 4..... A198920
... 3.... 4..... A198921
.. -1.... 1..... A198922
.. -1.... 2..... A198924
.. -1.... 3..... A198925
.. -1.... 4..... A198926
.. -2... -1..... A198927
.. -2.... 1..... A198928
.. -2.... 2..... A198929
.. -2.... 3..... A198930
.. -2.... 4..... A198931
.. -3... -1..... A198932
.. -3.... 1..... A198933
.. -3.... 2..... A198934
.. -3.... 4..... A198935
.. -4... -3..... A198936
.. -4... -2..... A198937
.. -4... -1..... A198938
.. -4.... 1..... A198939
.. -4.... 2..... A198940
.. -4.... 3..... A198941
.. -4.... 4..... A198942
... 1.... 2..... A198923
... 1.... 3..... A198983
... 1.... 4..... A198984
... 2.... 3..... A198985
... 3.... 4..... A198986
.. -1.... 1..... A198987
.. -1.... 2..... A198988
.. -1.... 3..... A198989
.. -1.... 4..... A198990
.. -2... -1..... A198991
.. -2.... 1..... A198992
.. -2... -3..... A198993
.. -3... -2..... A198994
.. -3... -1..... A198995
.. -2.... 1..... A198996
.. -3.... 2..... A198997
.. -3.... 3..... A198998
.. -3.... 4..... A198999
.. -4... -3..... A199000
.. -4... -1..... A199001
.. -4.... 1..... A199002
.. -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.

			1.32562251814753662348322902938798744330...

Index entries for transcendental numbers.

Cf. A197737, A198414.

(* 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 *)

A208672 a(n) = floor[1/(1-f(n))], where f(n) is the least nonnegative number such that f(n)^n = cos(f(n)).

Original entry on oeis.org

1, 3, 5, 7, 9, 10, 12, 14, 15, 17, 19, 20, 22, 23, 25, 27, 28, 30, 32, 33, 35, 37, 38, 40, 41, 43, 45, 46, 48, 50, 51, 53, 54, 56, 58, 59, 61, 63, 64, 66, 67, 69, 71, 72, 74, 76, 77, 79, 80, 82, 84, 85, 87, 89, 90, 92

Offset: 0

Ben Branman, Feb 29 2012

nonn

For n=0, the only possible solution is f(0)=0, which yields a(0)=1.
f(n)->1 as n->infinity.
a(n) ~ -n/log(cos(1))
f(1) = the Dottie number 0.73908513321516 = A003957
f(2) is A125578
f(3) is A125579
a(n) is defined for negative values of n as well.
If we let a(n)=floor[c(n)], c(n)=1/(1-f(n)), then f(n)^n=cos(f(n)) <=> 1-1/c(n) = cos(1-1/c(n))^(1/n) = exp(log(cos(1-1/c(n)))/n) = exp(log(cos(1)+O(1/c(n)^2))/n) = 1+log(cos(1))/n+o(1/n), assuming c(n) ~ c*n, which then yields c = -1/log(cos(1)). - M. F. Hasler, Mar 05 2012

			For n=4, the only positive solution to x^4=cos(x) is x=0.890553, so a(4)=floor(1/(1-.890553)) = floor(9.13682) = 9, so a(4)=9.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Dottie Number
Eric Weisstein's World of Mathematics, Cosine
Eric Weisstein's World of Mathematics, Fixed Point

Cf. A003957, A125578, A125579, A177413.

f[n_] := 1/(1 - FindRoot[x^n == Cos[x], {x, 0, 1}, WorkingPrecision -> 1000][[1,2]]); Table[Floor[f[n]], {n, 0, 100}]

a(n)=1\(1-solve(x=0,1,x^n-cos(x))) \\ Charles R Greathouse IV, Mar 04 2012

cp's OEIS Frontend

A197737 Decimal expansion of x<0 having x^2+x=cos(x).

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A198755 Decimal expansion of x>0 satisfying x^2+cos(x)=2.

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A208672 a(n) = floor[1/(1-f(n))], where f(n) is the least nonnegative number such that f(n)^n = cos(f(n)).

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