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 1-3 of 3 results.

A350162 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/(2*k-1))^2.

Original entry on oeis.org

1, 4, 8, 15, 25, 33, 45, 60, 73, 95, 115, 131, 157, 181, 205, 236, 270, 297, 333, 379, 403, 443, 487, 519, 578, 632, 672, 720, 778, 826, 886, 949, 989, 1059, 1131, 1186, 1260, 1332, 1388, 1482, 1564, 1612, 1696, 1776, 1858, 1946, 2038, 2102, 2187, 2308, 2380, 2490
Offset: 1

Views

Author

Seiichi Manyama, Dec 18 2021

Keywords

Links

Crossrefs

Column 2 of A350161.
Cf. A002654, A014200, A050469, A101455, A344720, A350143, A350166.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(k + 1) * Floor[n/(2*k - 1)]^2, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 18 2021 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^(k+1)*(n\(2*k-1))^2);
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, kronecker(-4, k/d)*(2*d-1)));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k/(1+x^(2*k)))/(1-x))

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} A101455(k/d) * (2*d - 1) = Sum_{k=1..n} 2 * A050469(k) - A002654(k) = 2 * A350166(n) - A014200(n).
G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k/(1 + x^(2*k)).

A350164 a(n) = Sum_{k=1..n}(-1)^(k+1) * floor(n/(2*k-1))^n.

Original entry on oeis.org

1, 4, 26, 255, 3125, 46593, 823415, 16776960, 387400807, 9999941975, 285311495511, 8916083675135, 302875039491581, 11112006557122561, 437893859877597389, 18446743921164642176, 827240261123526320144, 39346407973736968327497
Offset: 1

Views

Author

Seiichi Manyama, Dec 18 2021

Keywords

Crossrefs

Main diagonal of A350161.
Cf. A101455, A344724, A350145, A350167.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(k + 1) * Floor[n/(2*k - 1)]^n, {k, 1, n}]; Array[a, 18] (* Amiram Eldar, Dec 18 2021 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^(k+1)*(n\(2*k-1))^n);
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, kronecker(-4, k/d)*(d^n-(d-1)^n)));

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} A101455(k/d) * (d^n - (d - 1)^n).
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (k^n - (k - 1)^n) * x^k/(1 + x^(2*k)).
a(n) ~ n^n. - Vaclav Kotesovec, Dec 18 2021

A350163 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/(2*k-1))^3.

Original entry on oeis.org

1, 8, 26, 63, 125, 209, 335, 504, 703, 981, 1311, 1671, 2141, 2681, 3269, 3990, 4808, 5643, 6669, 7847, 8963, 10343, 11861, 13349, 15212, 17170, 19078, 21310, 23748, 26172, 28962, 31939, 34759, 38133, 41769, 45190, 49188, 53400, 57396, 62246, 67168, 71704, 77122
Offset: 1

Views

Author

Seiichi Manyama, Dec 18 2021

Keywords

Crossrefs

Column 3 of A350161.
Cf. A101455, A344721, A350144.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(k + 1) * Floor[n/(2*k - 1)]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 18 2021 *)
  • PARI
    a(n) = sum(k=1, n, (-1)^(k+1)*(n\(2*k-1))^3);
  • PARI
    a(n) = sum(k=1, n, sumdiv(k, d, kronecker(-4, k/d)*(d^3-(d-1)^3)));
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (k^3-(k-1)^3)*x^k/(1+x^(2*k)))/(1-x))

Formula

a(n) = Sum_{k=1..n} Sum_{d|k} A101455(k/d) * (d^3 - (d - 1)^3).
G.f.: (1/(1 - x)) * Sum_{k>=1} (k^3 - (k-1)^3) * x^k/(1 + x^(2*k)).
Showing 1-3 of 3 results.

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