A200111 -id:A200111 - cp's OEIS frontend

A199949 Decimal expansion of least x satisfying x^2 + cos(x) = 2*sin(x).

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

Offset: 0

Clark Kimberling, Nov 12 2011

For many choices of a,b,c, there are exactly two numbers x>0 satisfying a*x^2+b*cos(x)=c*sin(x).
Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... least x, greatest x
... 1.... 2.... A199949, A199950
... 1.... 3.... A199951, A199952
... 1.... 4.... A199953, A199954
... 2.... 3.... A199955, A199956
... 2.... 4.... A199957, A199958
... 3.... 3.... A199959, A199960
... 3.... 4.... A199961, A199962
... 4.... 3.... A199963, A199964
... 4.... 4.... A199965, A199966
... 1.... 3.... A199967, A200003
... 1.... 4.... A200004, A200005
... 1.... 4.... A200006, A200007
... 1.... 4.... A200008, A200009
.. -1.... 1.... A200010, A200011
.. -1.... 2.... A200012, A200013
.. -1.... 3.... A200014, A200015
.. -1.... 4.... A200016, A200017
.. -2.... 1.... A200018, A200019
.. -2.... 2.... A200020, A200021
.. -2.... 3.... A200022, A200023
.. -2.... 4.... A200024, A200025
.. -3.... 1.... A200026, A200027
.. -3.... 2.... A200093, A200094
.. -3.... 3.... A200095, A200096
.. -3.... 4.... A200097, A200098
.. -4.... 1.... A200099, A200100
.. -4.... 2.... A200101, A200102
.. -4.... 3.... A200103, A200104
.. -4.... 4.... A200105, A200106
.. -1.... 1.... A200107, A200108
.. -1.... 2.... A200109, A200110
.. -1.... 3.... A200111, A200112
.. -1.... 4.... A200114, A200115
.. -2.... 1.... A200116, A200117
.. -2.... 3.... A200118, A200119
.. -3.... 1.... A200120, A200121
.. -3.... 2.... A200122, A200123
.. -3.... 3.... A200124, A200125
.. -3.... 4.... A200126, A200127
.. -4.... 1.... A200128, A200129
.. -4.... 3.... A200130, A200131
.. -1.... 1.... A200132, A200133
.. -1.... 2.... A200223, A200224
.. -1.... 3.... A200225, A200226
.. -1.... 4.... A200227, A200228
.. -2.... 1.... A200229, A200230
.. -2.... 2.... A200231, A200232
.. -2.... 3.... A200233, A200234
.. -2.... 4.... A200235, A200236
.. -3.... 1.... A200237, A200238
.. -3.... 2.... A200239, A200240
.. -3.... 4.... A200241, A200242
.. -4.... 1.... A200277, A200278
.. -4.... 2.... A200279, A200280
.. -4.... 3.... A200281, A200282
.. -4.... 4.... A200283, A200284
.. -1.... 1.... A200285, A200286
.. -1.... 2.... A200287, A200288
.. -1.... 3.... A200289, A200290
.. -1.... 4.... A200291, A200292
.. -2.... 1.... A200293, A200294
.. -2.... 3.... A200295, A200296
.. -3.... 1.... A200299, A200300
.. -3.... 2.... A200297, A200298
.. -3.... 3.... A200301, A200302
.. -3.... 4.... A200303, A200304
.. -4.... 1.... A200305, A200306
.. -4.... 3.... A200307, A200308
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 A199949, take f(x,u,v)=x^2+u*cos(x)-v*sin(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.

			least x:  0.659266045766946074537348579563067611...
greatest x: 1.2710268008159460640047188480978502...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A199950.

(* Program 1:  A199949 *)
a = 1; b = 1; c = 2;
f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -1, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .65, .66}, WorkingPrecision -> 110]
RealDigits[r]   (* A199949 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.27, 1.28}, WorkingPrecision -> 110]
RealDigits[r]   (* A199950 *)
(* Program 2: implicit surface of x^2+u*cos(x)=v*sin(x) *)
f[{x_, u_, v_}] := x^2 + u*Cos[x] - v*Sin[x];
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, 1}]}, {u, -5, 0}, {v, 0, 1}];
ListPlot3D[Flatten[t, 1]]  (* for A199949 *)

a=1; b=1; c=2; solve(x=0, 1, a*x^2 + b*cos(x) - c*sin(x)) \\ G. C. Greubel, Jul 05 2018

A-number corrected by Jaroslav Krizek, Nov 27 2011

A200112 Decimal expansion of greatest x satisfying 2x^2-cos(x) = 3sin(x).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Nov 13 2011

See A199949 for a guide to related sequences. The Mathematica program includes a graph.

			least x: -0.27418592805983157901293857616592610671...
greatest x: 1.25741142949475925602237309814803895...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers.

Cf. A199949.

a = 2; b = -1; c = 3;
f[x_] := a*x^2 + b*Cos[x]; g[x_] := c*Sin[x]
Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -.28, -.27}, WorkingPrecision -> 110]
RealDigits[r]  (* A200111 *)
r = x /. FindRoot[f[x] == g[x], {x, 1.25, 1.26}, WorkingPrecision -> 110]
RealDigits[r]  (* A200112 *)

a=2; b=-1; c=3; solve(x=1.25, 1.26, a*x^2 + b*cos(x) - c*sin(x)) \\ G. C. Greubel, Jun 22 2018

A239100 Solution to the problem of finding the number of comparisons needed for optimal merging of 3 elements with n elements.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 6, 8, 10, 13, 17, 22, 28, 37, 47, 59, 75, 96, 120, 153, 194, 242, 309, 391, 487, 619, 784, 976, 1241, 1570, 1954, 2485, 3143, 3911, 4971, 6288, 7824, 9945, 12578, 15650, 19893, 25159, 31303, 39787, 50320, 62608, 79577, 100642, 125218, 159157

Offset: 1

N. J. A. Sloane, Mar 24 2014

nonn

F. K. Hwang, Optimal merging of 3 elements with n elements, SIAM J. Comput. 9 (1980), no. 2, 298--320. MR0568816 (82c:68022).
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,0,1,-1,0,2,-2).

Cf. A200310, A200111.

a(n) = if (n<9, v=[0, 1, 1, 2, 3, 4, 6, 8]; v[n], ndt = n\3; nmt = n%3; if (nmt==0, 43*2^(ndt-2)\7 - 2, if (nmt == 1, 107*2^(ndt-3)\7 - 2, (17*2^ndt-6)\7 - 1))); \\ Michel Marcus, Mar 26 2014

def A239100(n):
    if n <= 8: return (0,1,1,2,3,4,6,8)[n-1]
    r, b = divmod(n,3)
    return ((107<Chai Wah Wu, Mar 28 2023

For r >= 3, a(3r) = floor(43*2^(r-2)/7)-2,
a(3r+1) = floor(107*2^(r-3)/7)-2,
a(3r+2) = floor((17*2^r-6)/7)-1; initial terms are shown in sequence.
From Chai Wah Wu, Mar 28 2023: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-6) - a(n-7) + 2*a(n-9) - 2*a(n-10) for n > 18.
G.f.: x*(2*x^17 - x^16 - x^15 + x^14 + x^13 - x^12 + 2*x^11 - x^10 + x^8 + x^6 + x^5 + x^3 + x)/((x - 1)*(2*x^3 - 1)*(x^6 + x^3 + 1)). (End)

More terms from Michel Marcus, Mar 26 2014

cp's OEIS Frontend

A199949 Decimal expansion of least x satisfying x^2 + cos(x) = 2*sin(x).

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A200112 Decimal expansion of greatest x satisfying 2x^2-cos(x) = 3sin(x).

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A239100 Solution to the problem of finding the number of comparisons needed for optimal merging of 3 elements with n elements.

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Author

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PARI

Python

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cp's OEIS Frontend

A199949 Decimal expansion of least x satisfying x^2 + cos(x) = 2*sin(x).

Views

Author

Keywords

Comments

Examples

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Crossrefs

Programs

Mathematica

PARI

Extensions

A200112 Decimal expansion of greatest x satisfying 2*x^2-cos(x) = 3*sin(x).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A239100 Solution to the problem of finding the number of comparisons needed for optimal merging of 3 elements with n elements.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Python

Formula

Extensions

A200112 Decimal expansion of greatest x satisfying 2x^2-cos(x) = 3sin(x).