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 1-7 of 7 results.

A341697 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, 4, 4, 6, 7, 11, 11, 17, 17, 25, 29, 38, 38, 54, 54, 72, 80, 102, 102, 136, 140, 174, 186, 228, 228, 300, 300, 366, 388, 464, 480, 594, 594, 702, 736, 874, 874, 1068, 1068, 1250, 1324, 1528, 1528, 1828, 1844, 2144, 2220, 2534, 2534, 2982, 3026, 3464, 3572, 4028, 4028
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 17 2021

Keywords

Crossrefs

Cf. A038044, A122698, A333051, A341698.

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, 60}]
  • PARI
    A341697(n) = if(n<3, 1, sumdiv(n-1,d,if(d<(n-1), A341697((n-1)/d)*A341697(d), 0))); \\ Antti Karttunen, Feb 17 2021

A339755 a(1) = 1; a(n+1) = 1 + Sum_{d|n} a(n/d) * a(d).

Original entry on oeis.org

1, 2, 5, 11, 27, 55, 131, 263, 571, 1168, 2445, 4891, 10113, 20227, 40979, 82229, 165632, 331265, 665365, 1330731, 2666729, 5334769, 10679319, 21358639, 42740683, 85482096, 171004645, 342015001, 684113793, 1368227587, 2736633741, 5473267483, 10946869669, 21893763789, 43788190107
Offset: 1

Views

Author

Ilya Gutkovskiy, Dec 15 2020

Keywords

Crossrefs

Cf. A038044, A068336, A097558, A122698, A277120, A325303.

Programs

  • Maple
    a:= proc(n) option remember; uses numtheory;
      1+add(a(d)*a((n-1)/d), d=divisors(n-1))
    end:
seq(a(n), n=1..35);  # Alois P. Heinz, Dec 15 2020
  • Mathematica
    a[1] = 1; a[n_] := a[n] = 1 + Sum[a[(n - 1)/d] a[d], {d, Divisors[n - 1]}]; Table[a[n], {n, 1, 35}]

Formula

G.f.: x * (1/(1 - x) + Sum_{i>=1} Sum_{j>=1} a(i) * a(j) * x^(i*j)).
a(n) ~ c * 2^n, where c = 1.27442410710035207761153205319824525254716841098942446508584158048310907298... - Vaclav Kotesovec, Dec 16 2020

A325303 a(1) = 1; a(n+1) = -Sum_{d|n} a(n/d) * a(d).

Original entry on oeis.org

1, -1, 2, -4, 7, -14, 32, -64, 120, -244, 502, -1004, 1996, -3992, 8048, -16124, 32104, -64208, 128712, -257424, 514416, -1028960, 2058924, -4117848, 8233832, -16467713, 32939418, -65879316, 131750904, -263501808, 527020884, -1054041768, 2108050776, -4216103560, 8432271328
Offset: 1

Views

Author

Ilya Gutkovskiy, Sep 05 2019

Keywords

Links

Crossrefs

Cf. A038044, A122698.

Programs

  • Mathematica
    a[n_] := a[n] = -Sum[a[(n - 1)/d] a[d], {d, Divisors[n - 1]}]; a[1] = 1; Table[a[n], {n, 1, 35}]
  • PARI
    seq(n)={my(v=vector(n)); v[1]=1; for(n=1, #v-1, v[n+1] = -sumdiv(n, d, v[d]*v[n/d])); v} \\ Andrew Howroyd, Sep 05 2019

Formula

a(n) ~ -(-1)^n * c * 2^n, where c = 0.245410823583396667908354210407104718986708517177206856531763635090205896729... - Vaclav Kotesovec, Sep 09 2019

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}]

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).

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.
Showing 1-7 of 7 results.

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