cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A201741 Decimal expansion of the number x satisfying x^2+2=e^x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 04 2011

Keywords

Comments

For some choices of a, b, c, there is a unique value of x satisfying a*x^2+b*x+c=e^x, for other choices, there are two solutions, and for others, three. Guide to related sequences, with graphs included in Mathematica programs:
a.... b.... c.... x
1.... 0.... 2.... A201741
1.... 0.... 3.... A201742
1.... 0.... 4.... A201743
1.... 0.... 5.... A201744
1.... 0.... 6.... A201745
1.... 0.... 7.... A201746
1.... 0.... 8.... A201747
1.... 0.... 9.... A201748
1.... 0.... 10... A201749
-1... 0.... 1.... A201750, (x=0)
-1... 0.... 2.... A201751, A201752
-1... 0.... 3.... A201753, A201754
-1... 0.... 4.... A201755, A201756
-1... 0.... 5.... A201757, A201758
-1... 0.... 6.... A201759, A201760
-1... 0.... 7.... A201761, A201762
-1... 0.... 8.... A201763, A201764
-1... 0.... 9.... A201765, A201766
-1... 0.... 10... A201767, A201768
1.... 1.... 0.... A201769
1.... 1.... 1.... ..(x=0), A201770
1.... 1.... 2.... A201396
1.... 1.... 3.... A201562
1.... 1.... 4.... A201772
1.... 1.... 5.... A201889
1.... 2.... 1.... ..(x=0), A201890
1.... 2.... 2.... A201891
1.... 2.... 3.... A201892
1.... 2.... 4.... A201893
1.... 2.... 5.... A201894
1.... 3.... 1.... A201895, ..(x=0), A201896
1.... 3.... 2.... A201897, A201898, A201899
1.... 3.... 3.... A201900
1.... 3.... 4.... A201901
1.... 3.... 5.... A201902
1.... 4.... 1.... A201903, A201904
1.... 4.... 2.... A201905, A201906, A201907
1.... 4.... 3.... A201924, A201925, A201926
1.... 4.... 4.... A201927, A201928, A201929
1.... 4.... 5.... A201930
1.... 5.... 1.... A201931, A201932
1.... 5.... 2.... A201933, A201934, A201935
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 A201741, take f(x,u,v)=u*x^2+v-e^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.

Examples

			x=1.31907367685736535441789910952084846442196...

Links

Crossrefs

Cf. A201936.

Programs

  • Mathematica
    (* Program 1:  A201741 *)
a = 1; b = 0; c = 2;
f[x_] := a*x^2 + b*x + c; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -3, 3}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r]   (* A201741 *)
(* Program 2: implicit surface of u*x^2+v=E^x *)
f[{x_, u_, v_}] := u*x^2 + v - E^x;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 1, 5}]},
{v, 1, 3}, {u, 0, 5}];
ListPlot3D[Flatten[t, 1]] (* for A201741 *)

A346269 Expansion of e.g.f. 1/(2 - x^2 - exp(x)).

Original entry on oeis.org

1, 1, 5, 25, 195, 1781, 20043, 260317, 3881083, 64978861, 1209674883, 24764370533, 553130762451, 13383468009445, 348741065652619, 9736370899180813, 289948812396124875, 9174320178178480829, 307362076657095903411, 10869452423023391315413, 404614540610985119535715
Offset: 0

Views

Author

Vaclav Kotesovec, Jul 12 2021

Keywords

Links

Crossrefs

Cf. A006155, A201752.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Normal[Series[1/(2-x^2-E^x), {x, 0, nmax}]], x] * Range[0, nmax]!
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(2 - x^2 - exp(x)))) \\ Michel Marcus, Jul 12 2021
  • PARI
    b(n, m) = if(n==0, 1, sum(k=1, n, (1+(k==m)*m!)*binomial(n, k)*b(n-k, m)));
a(n) = b(n, 2); \\ Seiichi Manyama, Mar 12 2022

Formula

E.g.f.: 1/(2 - x^2 - exp(x)).
a(n) ~ n! / ((2 + 2*r - r^2) * r^(n+1)), where r = A201752 = 0.5372744491738566... is the positive root of the equation 2 - r^2 - exp(r) = 0.
a(0) = a(1) = 1; a(n) = n * (n-1) * a(n-2) + Sum_{k=1..n} binomial(n,k) * a(n-k). - Seiichi Manyama, Mar 11 2022

A201751 Decimal expansion of the least x satisfying -x^2+2=e^x.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 05 2011

Keywords

Comments

See A201741 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			least:  -1.3159737777962901878871773873012710...
greatest:  0.53727444917385660425676298977967...

Links

Crossrefs

Cf. A201741.

Programs

  • Mathematica
    a = -1; b = 0; c = 2;
f[x_] := a*x^2 + b*x + c; g[x_] := E^x
Plot[{f[x], g[x]}, {x, -2, 1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, -1.4, -1.3}, WorkingPrecision -> 110]
RealDigits[r]    (* A201751 *)
r = x /. FindRoot[f[x] == g[x], {x, .5, .6}, WorkingPrecision -> 110]
RealDigits[r]    (* A201752 *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.