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.

Previous Showing 11-17 of 17 results.

A333051 a(1) = 1; a(n+1) = Sum_{d|n, gcd(d, n/d) = 1} a(n/d) * a(d).

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 36, 72, 144, 288, 592, 1184, 2384, 4768, 9608, 19248, 38496, 76992, 154272, 308544, 617152, 1234448, 2470080, 4940160, 9880608, 19761216, 39527200, 79054400, 158109088, 316218176, 632456976, 1264913952, 2529827904, 5059658176, 10119393344, 20238787264
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 06 2020

Keywords

Links

Crossrefs

Cf. A038044, A122698.

Programs

  • Maple
    a[1]:= 1:
for n from 1 to 40 do
  P:= ifactors(n)[2];
  k:= nops(P);
  t:= 0;
  for S in combinat:-powerset(k) do
    d:= mul(P[i][1]^P[i][2],i=S);
    t:= t + a[d]*a[n/d]
  od;
  a[n+1]:= t
od:
seq(a[i],i=1..41); # Robert Israel, Mar 09 2020
  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[If[GCD[(n - 1)/d, d] == 1, a[(n - 1)/d] a[d], 0], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 36}]

A341698 a(1) = a(2) = 1; a(n+1) = -Sum_{d|n, d < n} a(n/d) * a(d).

Original entry on oeis.org

1, 1, -1, 1, -2, 2, 0, 0, -2, 1, 3, -3, 1, -1, 1, -5, 4, -4, 12, -12, 14, -14, 8, -8, 10, -14, 12, -16, 18, -18, 26, -26, 36, -30, 22, -22, 24, -24, 0, 2, 20, -20, -10, 10, 12, -18, 2, -2, 14, -14, -2, 10, 16, -16, -8, 20, 14, 10, -46, 46, -52, 52, -104, 132, -70, 74, -186, 186, -134, 150
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 17 2021

Keywords

Crossrefs

Cf. A038044, A325303, A341697.

Programs

  • Mathematica
    a[1] = a[2] = 1; a[n_] := a[n] = -Sum[If[d < (n - 1), a[(n - 1)/d] a[d], 0], {d, Divisors[n - 1]}]; Table[a[n], {n, 70}]
  • PARI
    A341698(n) = if(n<3, 1, sumdiv(n-1,d,if(d<(n-1), -A341698((n-1)/d)*A341698(d), 0))); \\ Antti Karttunen, Feb 17 2021

A351787 a(1) = 1; a(n+1) = a(n) + Sum_{d|n} a(n/d) * a(d).

Original entry on oeis.org

1, 2, 6, 18, 58, 174, 546, 1638, 4986, 14994, 45214, 135642, 407838, 1223514, 3672726, 11018874, 33063498, 99190494, 297593514, 892780542, 2678403690, 8035217622, 24105833722, 72317501166, 216953071986, 650859219322, 1952579289318, 5857737927786, 17573218697070
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 19 2022

Keywords

Crossrefs

Cf. A038044, A084978, A122698, A277120, A339755, A341697, A345139, A351788.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[(n - 1)/d] a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 29}]

Formula

G.f.: x * ( 1 + Sum_{i>=1} Sum_{j>=1} a(i) * a(j) * x^(i*j) ) / (1 - x).

A351788 a(1) = 1; a(n) = a(n-1) + Sum_{d|n, 1 < d < n} a(n/d) * a(d).

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 8, 9, 13, 13, 25, 25, 33, 37, 57, 57, 83, 83, 117, 125, 151, 151, 233, 237, 287, 305, 387, 387, 503, 503, 649, 675, 789, 805, 1073, 1073, 1239, 1289, 1607, 1607, 1955, 1955, 2309, 2419, 2721, 2721, 3465, 3481, 4007, 4121, 4795, 4795, 5643, 5695
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 19 2022

Keywords

Crossrefs

Cf. A038044, A084978, A122698, A277120, A339755, A341697, A345139, A351787.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = a[n - 1] + Sum[If[1 < d < n, a[n/d] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 55}]

Formula

G.f.: ( x + Sum_{i>=2} Sum_{j>=2} a(i) * a(j) * x^(i*j) ) / (1 - x).

A307816 a(1) = 1; a(n) = Sum_{k=1..n-1} a(n-k) * Sum_{d|k} a(d)*a(k/d).

Original entry on oeis.org

1, 1, 3, 11, 46, 201, 928, 4399, 21431, 106399, 536896, 2744532, 14185314, 73999955, 389131156, 2060478226, 10976863244, 58792036053, 316397505099, 1710037259744, 9277953713444, 50514377326158, 275903656802218, 1511334791637679, 8300811367229306, 45703063861360901
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 30 2019

Keywords

Crossrefs

Cf. A038044, A317875.

Programs

  • Mathematica
    a[n_] := a[n] = Sum[a[n - k] Sum[a[d] a[k/d], {d, Divisors[k]}], {k, 1, n - 1}]; a[1] = 1; Table[a[n], {n, 1, 26}]
a[n_] := a[n] = SeriesCoefficient[x + Sum[a[k] x^k, {k, 1, n - 1}]  Sum[Sum[a[i] a[j] x^(i j), {j, 1, n - 1}], {i, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 26}]
  • PARI
    lista(nn) = { my(va=vector(nn)); va[1] = 1; for (n=2, nn, va[n] = sum(k=1, n-1, va[n-k] * sumdiv(k, d, va[d]*va[k/d]))); va;} \\ Michel Marcus, Apr 30 2019

Formula

G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x + (Sum_{n>=1} a(n)*x^n) * (Sum_{i>=1} Sum_{j>=1} a(i)*a(j)*x^(i*j)).

A341639 a(1) = 1; a(n+1) = Sum_{d|n} phi(d) * a(d) * a(n/d).

Original entry on oeis.org

1, 1, 2, 6, 19, 95, 291, 2037, 10203, 71429, 357240, 3929640, 19648533, 255430929, 1788018540, 16092167088, 144829514049, 2462101738833, 17234712244012, 327459532636228, 2947135794083881, 38312765323095109, 421440418557975839, 9693129626833444297, 87238166641520673597
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 16 2021

Keywords

Crossrefs

Cf. A000010, A038044, A038045, A332791.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = Sum[EulerPhi[d] a[d] a[(n - 1)/d], {d, Divisors[n - 1]}]; Table[a[n], {n, 25}]
a[1] = 1; a[n_] := a[n] = Sum[a[GCD[n - 1, k]] a[(n - 1)/GCD[n - 1, k]], {k, n - 1}]; Table[a[n], {n, 25}]

Formula

a(1) = 1; a(n+1) = Sum_{k=1..n} a(gcd(n,k)) * a(n/gcd(n,k)).

A351797 a(1) = 1; a(n+1) = -a(n) + 2 * Sum_{d|n} a(n/d) * a(d).

Original entry on oeis.org

1, 1, 3, 9, 29, 87, 273, 819, 2493, 7497, 22607, 67821, 203919, 611757, 1836363, 5509437, 16531749, 49595247, 148796757, 446390271, 1339201845, 4017608811, 12052916861, 36158750583, 108476535993, 325429609661, 976289644659, 2928868963893, 8786609348535
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 19 2022

Keywords

Crossrefs

Cf. A038044, A084978, A122698, A277120, A339755, A341697, A345139, A351787, A351788.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = -a[n - 1] + 2 Sum[a[(n - 1)/d] a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 29}]

Formula

G.f.: x * ( 1 + 2 * Sum_{i>=1} Sum_{j>=2} a(i) * a(j) * x^(i*j) ) / (1 - x).
a(n) = A351787(n) / 2 for n > 1.
Previous Showing 11-17 of 17 results.

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