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 24 results. Next

A226571 Decimal expansion of lim_{k->oo} f(k), where f(1)=2, and f(k) = 2 - log(f(k-1)) for k>1.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jun 11 2013

Keywords

Comments

Old definition was: Decimal digits of limit(f(n)), where f(1) = 2 - log(2), f(n) = f(f(n-1)).
Let h(x) be lesser of the two solutions of s - log(s) = x; then A226571 represents h(2). The function h(x) is plotted by the Mathematica program. [This comment is wrong. A226571 = 1.5571455989976... is the unique root of the equation s + log(s) = 2. Equation s - log(s) = 2 does have two roots, but they are s = 3.14619322062... (=A226572) and s = 0.158594339563... (not A226571). - Vaclav Kotesovec, Jan 09 2014]

Examples

			2 - log 2 = 1.732378...
2 - log(2 - log 2) = 1.450504...
2 - log(2 - log(2 - log 2)) = 1.628088...
limit(f(n)) = 1.557144510523...

Links

Crossrefs

Cf. A006155, A226572, A226573, A226574.

Programs

  • Mathematica
    f[s_, accuracy_] := FixedPoint[N[s - Log[#], accuracy] &, 1]
g[s_, accuracy_] := FixedPoint[N[s + Log[#], accuracy] &, 1]
d1 = RealDigits[f[2, 200]][[1]]  (* A226571 *)
d2 = RealDigits[g[2, 200]][[1]]  (* A226572 *)
s /. NSolve[s - Log[s] == 2, 200]  (* both constants *)
h[x_] := s /. NSolve[s - Log[s] == x] Plot[h[x], {x, 1, 3}, PlotRange -> {0, 1}] (* bottom branch of h *)
Plot[h[x], {x, 1, 3}, PlotRange -> {1, 5}] (* top branch *)
RealDigits[LambertW[Exp[2]], 10, 50][[1]] (* G. C. Greubel, Nov 14 2017 *)
  • PARI
    lambertw(exp(2)) \\ G. C. Greubel, Nov 14 2017

Formula

Equals LambertW(exp(2)). - Vaclav Kotesovec, Jan 09 2014

Extensions

Definition edited by N. J. A. Sloane, Dec 09 2017

A367835 Expansion of e.g.f. 1/(2 - x - exp(2*x)).

Original entry on oeis.org

1, 3, 22, 242, 3544, 64872, 1424976, 36517840, 1069533824, 35240047232, 1290137297152, 51955085596416, 2282489348834304, 108630445541684224, 5567741266098944000, 305752314499878569984, 17909736027185859100672, 1114647522476340562132992
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Cf. A006155, A367836, A367837.
Cf. A126390, A216794, A343672, A355110, A367838.

Programs

  • Maple
    A367835 := proc(n)
    option remember ;
    if n = 0 then
        1 ;
    else
        n*procname(n-1)+add(2^k*binomial(n,k)*procname(n-k),k=1..n) ;
    end if;
end proc:
seq(A367835(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 2^j*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 2^k * binomial(n,k) * a(n-k).

A355110 Expansion of e.g.f. 2 / (3 - 2*x - exp(2*x)).

Original entry on oeis.org

1, 2, 10, 76, 768, 9696, 146896, 2596448, 52449536, 1191944704, 30097334784, 835973778432, 25330620762112, 831497823494144, 29394162040580096, 1113330929935101952, 44979662118902366208, 1930798895281527717888, 87756941394038739828736, 4210241529540625311727616
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 19 2022

Keywords

Crossrefs

Cf. A004123, A006155, A007405, A122704, A343672, A355111, A355112, A355113, A355114.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[2/(3 - 2 x - Exp[2 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 2^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 2^(k-1) * a(n-k).
a(n) ~ n! / ((1 + LambertW(exp(3))) * ((3 - LambertW(exp(3)))/2)^(n+1)). - Vaclav Kotesovec, Jun 19 2022

A355111 Expansion of e.g.f. 3 / (4 - 3*x - exp(3*x)).

Original entry on oeis.org

1, 2, 11, 93, 1041, 14541, 243747, 4767183, 106556373, 2679469065, 74864397015, 2300883358995, 77144051804409, 2802027511061325, 109604157405491691, 4593512301562215783, 205348466229473678301, 9753645833118762303249, 490530576727430107027839, 26040317900991310393061499
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 19 2022

Keywords

Crossrefs

Cf. A003575, A006155, A032033, A255927, A343673, A355110, A355112, A355113, A355114.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[3/(4 - 3 x - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 3^(k-1) * a(n-k).
a(n) ~ n! / ((1 + LambertW(exp(4))) * ((4 - LambertW(exp(4)))/3)^(n+1)). - Vaclav Kotesovec, Jun 19 2022

A355112 Expansion of e.g.f. 4 / (5 - 4*x - exp(4*x)).

Original entry on oeis.org

1, 2, 12, 112, 1376, 21056, 386688, 8286720, 202958848, 5592199168, 171203895296, 5765504860160, 211811563929600, 8429932686999552, 361312700788375552, 16592261047219388416, 812749365813312487424, 42299637489384965537792, 2330989060564353634271232
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 19 2022

Keywords

Crossrefs

Cf. A003576, A006155, A094417, A326324, A343674, A355110, A355111, A355113, A355114.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[4/(5 - 4 x - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = n a[n - 1] + Sum[Binomial[n, k] 4^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} binomial(n,k) * 4^(k-1) * a(n-k).
a(n) ~ n! / ((1 + LambertW(exp(5))) * ((5 - LambertW(exp(5)))/4)^(n+1)). - Vaclav Kotesovec, Jun 19 2022

A367836 Expansion of e.g.f. 1/(2 - x - exp(3*x)).

Original entry on oeis.org

1, 4, 41, 627, 12759, 324543, 9906453, 352785933, 14358074211, 657405969075, 33444798498657, 1871613674744553, 114259520317835871, 7556674046930376111, 538212358684663414317, 41071433946325564954581, 3343141735414440335583003
Offset: 0

Views

Author

Seiichi Manyama, Dec 02 2023

Keywords

Crossrefs

Cf. A006155, A367835, A367837.
Cf. A284859, A343673, A355111.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(2-x-Exp[3x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 16 2024 *)
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i*v[i]+sum(j=1, i, 3^j*binomial(i, j)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = n * a(n-1) + Sum_{k=1..n} 3^k * binomial(n,k) * a(n-k).

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

A343672 a(0) = 1; a(n) = 2 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).

Original entry on oeis.org

1, 3, 19, 181, 2299, 36501, 695427, 15457709, 392672651, 11221959685, 356339728243, 12446649786429, 474273933636411, 19577992095770837, 870345573347448803, 41455153171478627533, 2106173029315813515883, 113694251997087087941925, 6498401704686168598548435, 392062852538564346207533789
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000670, A005493, A006155, A032032, A343673, A343674.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = 2 n a[n - 1] + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[1/(2 (1 - x) - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1 / (2 * (1 - x) - exp(x)).
a(n) ~ n! / (2*(1 + LambertW(exp(1)/2)) * (1 - LambertW(exp(1)/2))^(n+1)). - Vaclav Kotesovec, Jun 20 2022

A343673 a(0) = 1; a(n) = 3 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).

Original entry on oeis.org

1, 4, 33, 409, 6759, 139621, 3460989, 100091335, 3308146179, 123005753041, 5081871122073, 230948185830187, 11449697796242319, 614944043618257237, 35568197580789653685, 2204201734650777596863, 145703352769994600516187, 10233323176300508748808921, 761004837938469796089586257
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000670, A005494, A006155, A032032, A343672, A343674.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = 3 n a[n - 1] + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(2 - 3 x - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1 / (2 - 3*x - exp(x)).
a(n) ~ n! * 3^n / ((1 + LambertW(exp(2/3)/3)) * (2 - 3*LambertW(exp(2/3)/3))^(n+1)). - Vaclav Kotesovec, Jun 20 2022

A343674 a(0) = 1; a(n) = 4 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).

Original entry on oeis.org

1, 5, 51, 781, 15947, 407021, 12466251, 445452813, 18191122219, 835737327661, 42661645147403, 2395510523568845, 146739531459316587, 9737742346694258157, 695911661109898805323, 53286006304099668950413, 4352120920347139791200171, 377674509364714706139413933, 34702277449656625185428239755
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 25 2021

Keywords

Crossrefs

Cf. A000670, A006155, A032032, A045379, A343672, A343673.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = 4 n a[n - 1] + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(2 (1 - 2 x) - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

E.g.f.: 1 / (2 * (1 - 2*x) - exp(x)).
a(n) ~ n! * 2^(n-1) / ((1 + LambertW(exp(1/2)/4)) * (1 - 2*LambertW(exp(1/2)/4))^(n+1)). - Vaclav Kotesovec, Jun 20 2022
Showing 1-10 of 24 results. Next

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