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 1-5 of 5 results.

A144374 Triangle T(n,k), n>=1, 1<=k<=n, read by rows, where sequence a_k of column k begins with (k+1) 1's and a_k(n) shifts k places down under Dirichlet convolution.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 9, 2, 2, 1, 1, 18, 5, 2, 2, 1, 1, 40, 4, 3, 2, 2, 1, 1, 80, 12, 4, 3, 2, 2, 1, 1, 168, 8, 6, 2, 3, 2, 2, 1, 1, 340, 28, 6, 6, 2, 3, 2, 2, 1, 1, 698, 17, 10, 4, 4, 2, 3, 2, 2, 1, 1, 1396, 60, 13, 8, 4, 4, 2, 3, 2, 2, 1, 1, 2844, 34, 16, 5, 6, 2, 4, 2, 3, 2, 2, 1, 1, 5688
Offset: 1

Views

Author

Alois P. Heinz, Sep 18 2008

Keywords

Comments

Sequence a_k of column k begins with k terms from A000012 (only the last is in the triangle), followed by the first (k+1) terms from A000005.

Examples

			Triangle begins:
   1;
   1,  1;
   2,  1, 1;
   4,  2, 1, 1;
   9,  2, 2, 1, 1;
  18,  5, 2, 2, 1, 1;

Links

Crossrefs

Columns 1-9 give: A038044, A144366, A144367, A144368, A144369, A144370, A144371, A144372, A144373.
Cf. A000012, A000005.

Programs

  • Maple
    with(numtheory): dck:= proc(b,c) proc(n, k) option remember; add(b(d,k) *c(n/d,k), d=`if`(n<0,{}, divisors(n))) end end: B:= dck(T,T): T:= (n, k)-> if n<=k then 1 else B(n-k, k) fi: seq(seq(T(n, k), k=1..n), n=1..14);
  • Mathematica
    dck[b_, c_][n_, k_] := dck[b, c][n, k] = Sum[b[d, k]*c[n/d, k], {d, If[n < 0, {}, Divisors[n]]}]; B = dck[T, T]; T[n_, k_] := If[n <= k, 1, B[n-k, k]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 15 2014, translated from Maple *)

A307992 G.f. A(x) satisfies: A(x) = x + x^2 * (1 + A(x) + 2*A(x^2) + 3*A(x^3) + ...).

Original entry on oeis.org

1, 1, 1, 3, 4, 9, 9, 20, 16, 38, 28, 61, 39, 110, 52, 149, 84, 225, 101, 317, 120, 454, 175, 543, 198, 823, 243, 940, 327, 1259, 356, 1601, 387, 2051, 515, 2270, 623, 3114, 660, 3373, 829, 4381, 870, 5145, 913, 6264, 1245, 6683, 1292, 8776, 1404, 9477, 1724
Offset: 1

Views

Author

Ilya Gutkovskiy, May 09 2019

Keywords

Links

Crossrefs

Cf. A007439, A038046, A144366, A318372.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<3, signum(n), (m->
      m*add(a(d)/d, d=numtheory[divisors](m)))(n-2))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, May 09 2019
  • Mathematica
    terms = 57; A[] = 0; Do[A[x] = x + x^2 (1 + Sum[k A[x^k], {k, 1, terms}]) + O[x]^(terms + 1) // Normal, terms + 1]; Rest[CoefficientList[A[x], x]]
a[n_] := a[n] = SeriesCoefficient[x + x^2 (1 + Sum[a[k] x^k/(1 - x^k)^2, {k, 1, n - 1}]), {x, 0, n}]; Table[a[n], {n, 1, 57}]
a[n_] := a[n] = Sum[d a[(n - 2)/d], {d, Divisors[n - 2]}]; a[1] = a[2] = 1; Table[a[n], {n, 1, 57}]

Formula

G.f.: x + x^2 * (1 + Sum_{n>=1} a(n)*x^n/(1 - x^n)^2).
a(1) = a(2) = 1; a(n+2) = Sum_{d|n} d*a(n/d).

A368376 Arises from enumeration of a certain class of zig-zag knight's paths on the square grid.

Original entry on oeis.org

0, 1, 0, 1, 0, 3, 1, 6, 3, 13, 9, 29, 25, 65, 66, 148, 171, 341, 437, 793, 1107, 1860, 2790, 4395, 7009, 10452, 17574, 24999, 44019, 60097, 110210, 145130, 275925, 351916, 690993, 856502, 1731224, 2091599, 4339980, 5123437, 10887192, 12585354, 27331465
Offset: 0

Views

Author

N. J. A. Sloane, Feb 18 2024

Keywords

Comments

It would be nice to have a more precise definition.

Links

Crossrefs

Cf. A144366, A166297, A368377, A004148.
A093128 is a bisection.

Programs

  • Mathematica
    r = (1 - z^4 - z^2 - Sqrt[z^8 - 2z^6 - z^4 - 2z^2 + 1]) / (2z^3);
gf = r (u^2 z + u z^2 + 1) / (z^3 (1 - r u));
Table[SeriesCoefficient[gf,{u,0,2},{z,0,n}], {n,0,33}] (* Andrei Zabolotskii, Jul 25 2025 *)

Formula

G.f.: (x + x^2 * R(x) + R(x)^2) * R(x) / x^3, where R(x) = x * (A(x^2) - 1) and A(x) is the g.f. of A004148. - Andrei Zabolotskii, Jul 25 2025

Extensions

Terms a(14) and beyond from Andrei Zabolotskii, Jul 25 2025

A368377 Arises from enumeration of a certain class of zig-zag knight's paths on the square grid.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 0, 6, 1, 15, 4, 37, 14, 91, 44, 222, 129, 541, 364, 1319, 1000, 3219, 2696, 7869, 7172, 19273, 18892, 47299, 49398, 116317, 128444, 286624, 332552, 707679, 858168, 1750588, 2208898, 4338314, 5674380, 10769893, 14554398, 26780522, 37286820
Offset: 0

Views

Author

N. J. A. Sloane, Feb 18 2024

Keywords

Comments

It would be nice to have a more precise definition.

Links

Crossrefs

Cf. A144366, A368376.

Programs

  • Mathematica
    r = (1 - z^4 - z^2 - Sqrt[z^8 - 2z^6 - z^4 - 2z^2 + 1]) / (2z^3);
gf = r (u^2 z + u z^2 + 1) / (z^3 (1 - r u));
Table[SeriesCoefficient[gf,{u,0,3},{z,0,n}], {n,0,50}] (* Andrei Zabolotskii, Jul 25 2025 *)

Formula

G.f.: (x + x^2 * R(x) + R(x)^2) * R(x)^2 / x^3 = F(x) * R(x), where R(x) = x * (A(x^2) - 1), A(x) is the g.f. of A004148, and F(x) is the g.f. of A368376. - Andrei Zabolotskii, Jul 25 2025

Extensions

Terms a(16) and beyond from Andrei Zabolotskii, Jul 25 2025

A346116 a(1) = a(2) = 1; a(n+2) = 1 + Sum_{d|n} a(n/d) * a(d).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 11, 21, 23, 49, 51, 109, 103, 247, 207, 517, 435, 1086, 871, 2251, 1743, 4631, 3531, 9365, 7063, 19081, 14152, 38369, 28397, 77299, 56795, 155289, 113591, 311739, 227387, 624349, 454885, 1251509, 909771, 2504761, 1819955, 5014529, 3639911, 10033709, 7279823
Offset: 1

Views

Author

Ilya Gutkovskiy, Jul 05 2021

Keywords

Crossrefs

Cf. A144366, A277120, A339755.

Programs

  • Mathematica
    a[1] = a[2] = 1; a[n_] := a[n] = 1 + Sum[a[(n - 2)/d] a[d], {d, Divisors[n - 2]}]; Table[a[n], {n, 1, 45}]

Formula

G.f.: x + x^2 * (1/(1 - x) + Sum_{i>=1} Sum_{j>=1} a(i) * a(j) * x^(i*j)).
Showing 1-5 of 5 results.

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