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 1-10 of 10 results.

A201946 Decimal expansion of x>0 satisfying x*sinh(x)=2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 15 2011

Keywords

Comments

For many choices of u and v, there is exactly one x>0 satisfying x*sinh(u*x)=v. Guide to related sequences, with graphs included in Mathematica programs:
u.... v.... x
1.... 1.... A133867
1.... 2.... A201946
1.... 3.... A202243
2.... 1.... A202244
3.... 1.... A202245
2.... 2.... A202284
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 A199597, take f(x,u,v)=x*sinh(ux)-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.

Examples

			1.2493943366463244725112743212610081234694...

Links

Crossrefs

Cf. A201939.

Programs

  • Mathematica
    (* Program 1:  A201946 *)
u = 1; v = 2;
f[x_] := x*Sinh[u*x]; g[x_] := v
Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.2, 1.3}, WorkingPrecision -> 110]
RealDigits[r]   (* A201946 *)
(* Program 2: implicit surface of u*sinh(x)=v *)
f[{x_, u_, v_}] := x*Sinh[u*x] - v;
t = Table[{u, v, x /. FindRoot[f[{x, u, v}] == 0, {x, 0, .2}]}, {v, 0, 10}, {u, 1, 4}];
ListPlot3D[Flatten[t, 1]] (* for A201946 *)

A381280 Expansion of e.g.f. 1/(1 - x * cosh(2*x)).

Original entry on oeis.org

1, 1, 2, 18, 120, 920, 10320, 126448, 1714048, 27073152, 472354560, 8989147904, 187690331136, 4245706716160, 103239264593920, 2691918892861440, 74885151106498560, 2212607133043884032, 69227613551324233728, 2286465386258267176960, 79487593489348266557440
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2025

Keywords

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Crossrefs

Cf. A205571, A381281.
Cf. A185951, A201939.

Programs

  • PARI
    a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*2^(n-k)*a185951(n, k));

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} 4^k * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).
a(n) = Sum_{k=0..n} k! * 2^(n-k) * A185951(n,k).
a(n) ~ sqrt(Pi) * 2^(n + 5/2) * n^(n + 1/2) / ((1 + sinh(r))^2 * exp(n) * r^(n+2)), where r = A201939. - Vaclav Kotesovec, Apr 19 2025

A201943 Decimal expansion of x>0 satisfying x*cosh(x)=3.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 15 2011

Keywords

Comments

See A201939 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			1.3976589774224380204403252065308389839789...

Links

Crossrefs

Cf. A201939.

Programs

  • Mathematica
    u = 1; v = 3;
f[x_] := x*Cosh[u*x]; g[x_] := v
Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.3, 1.4}, WorkingPrecision -> 110]
RealDigits[r]    (* A201943 *)

A201944 Decimal expansion of x>0 satisfying x*cosh(2x)=1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 15 2011

Keywords

Comments

See A201939 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			0.5752924837093319747674668668068940978834187...

Links

Crossrefs

Cf. A201939.

Programs

  • Mathematica
    u = 2; v = 1;
f[x_] := x*Cosh[u*x]; g[x_] := v
Plot[{f[x], g[x]}, {x, 0, 1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .57, .58}, WorkingPrecision -> 110]
RealDigits[r]    (* A201944 *)

A201945 Decimal expansion of x>0 satisfying x*cosh(3x)=1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 15 2011

Keywords

Comments

See A201939 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			0.465886325807479340146775068843612994659633...

Links

Crossrefs

Cf. A201939.

Programs

  • Mathematica
    u = 3; v = 1;
f[x_] := x*Cosh[u*x]; g[x_] := v
Plot[{f[x], g[x]}, {x, 0, 1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .46, .47}, WorkingPrecision -> 110]
RealDigits[r]     (* A201945 *)

A202243 Decimal expansion of x>0 satisfying x*sinh(x)=3.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 15 2011

Keywords

Comments

See A201939 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			x=1.46487733436345412578929000375510637762949493...

Crossrefs

Cf. A201946.

Programs

  • Mathematica
    u = 1; v = 3;
f[x_] := x*Sinh[u*x]; g[x_] := v
Plot[{f[x], g[x]}, {x, 0, 2}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, 1.4, 1.5}, WorkingPrecision -> 110]
RealDigits[r]   (* A202243 *)

A202244 Decimal expansion of x>0 satisfying x*sinh(2x)=1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 15 2011

Keywords

Comments

See A201939 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			x=0.62469716832316223625563716063050406173...

Crossrefs

Cf. A201946.

Programs

  • Mathematica
    u = 2; v = 1;
f[x_] := x*Sinh[u*x]; g[x_] := v
Plot[{f[x], g[x]}, {x, 0, 1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .6, .7}, WorkingPrecision -> 110]
RealDigits[r]   (* A202244 *)

A202245 Decimal expansion of x>0 satisfying x*sinh(3x)=1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 15 2011

Keywords

Comments

See A201939 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			x=0.48829244478781804192976333458503545920...

Crossrefs

Cf. A201946.

Programs

  • Mathematica
    u = 3; v = 1;
f[x_] := x*Sinh[u*x]; g[x_] := v
Plot[{f[x], g[x]}, {x, 0, 1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .4, .5}, WorkingPrecision -> 110]
RealDigits[r]   (* A202245 *)

A202283 Decimal expansion of x>0 satisfying x*cosh(2x)=2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 15 2011

Keywords

Comments

See A201939 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			0.79015026139471643916176884854422171239618...

Links

Crossrefs

Cf. A201939.

Programs

  • Mathematica
    u = 2; v = 2;
f[x_] := x*Cosh[u*x]; g[x_] := v
Plot[{f[x], g[x]}, {x, 0, 1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .7, .8}, WorkingPrecision -> 110]
RealDigits[r]   (* A202283 *)

A202284 Decimal expansion of x>0 satisfying x*sinh(2x)=2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 15 2011

Keywords

Comments

See A201939 for a guide to related sequences. The Mathematica program includes a graph.

Examples

			x=0.81500182386698136500646474507749839416761248...

Crossrefs

Cf. A201946.

Programs

  • Mathematica
    u = 2; v = 2;
f[x_] := x*Sinh[u*x]; g[x_] := v
Plot[{f[x], g[x]}, {x, 0, 1}, {AxesOrigin -> {0, 0}}]
r = x /. FindRoot[f[x] == g[x], {x, .8, .9}, WorkingPrecision -> 110]
RealDigits[r]   (* A202284 *)
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