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.

A229653 Trisection a(3n+k) gives k-th differences of a for k=0..2 with a(n)=0 for n<2 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 0, 1, -2, 1, -1, 2, 0, 1, -4, 1, -3, 6, -2, 3, -5, 1, -2, 5, -1, 3, -5, 2, -2, 3, 0, 1, -6, 1, -5, 10, -4, 5, -9, 1, -4, 13, -3, 9, -17, 6, -8, 13, -2, 5, -13, 3, -8, 14, -5, 6, -9, 1, -3, 10, -2, 7, -13, 5, -6, 10, -1, 4, -12, 3, -8, 15, -5, 7, -11
Offset: 0

Views

Author

Alois P. Heinz, Sep 27 2013

Keywords

Links

Crossrefs

Cf. A005590, A229654, A229655, A229656, A229657, A229658, A229659, A229660.

Programs

  • Maple
    a:= proc(n) option remember; local m, q;
      m:= irem(n, 3, 'q'); `if`(n<3, `if`(n=2, 1, 0),
      add(a(q+m-j)*(-1)^j*binomial(m, j), j=0..m))
    end:
seq(a(n), n=0..100);
  • Mathematica
    a[n_] := a[n] = Module[{m, q}, {q, m} = QuotientRemainder[n, 3]; If[n < 3, If[n == 2, 1, 0], Sum[a[q + m - j]*(-1)^j*Binomial[m, j], {j, 0, m}]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 09 2018, translated from Maple *)

Formula

a(3*n) = a(n),
a(3*n+1) = a(n+1) - a(n),
a(3*n+2) = a(n+2) - 2*a(n+1) + a(n).

A229654 Quadrisection a(4n+k) gives k-th differences of a for k=0..3 with a(n)=0 for n<3 and a(3)=1.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, -3, 0, 1, -2, 3, 1, -1, 1, 0, 0, 0, 1, -6, 0, 1, -5, 12, 1, -4, 7, -9, -3, 3, -2, -2, 0, 1, -4, 12, 1, -3, 8, -15, -2, 5, -7, 7, 3, -2, 0, 4, 1, -2, 4, -7, -1, 2, -3, 4, 1, -1, 1, -1, 0, 0, 0, 1, 0, 0, 1, -9, 0, 1, -8, 21, 1, -7, 13, -18
Offset: 0

Views

Author

Alois P. Heinz, Sep 27 2013

Keywords

Links

Crossrefs

Cf. A005590, A229653, A229655, A229656, A229657, A229658, A229659, A229660.

Programs

  • Maple
    a:= proc(n) option remember; (m-> `if`(n<4, `if`(n=3, 1, 0), add(
       a(q+m-j)*(-1)^j*binomial(m, j), j=0..m)))(irem(n, 4, 'q'))
    end:
seq(a(n), n=0..100);
  • Mathematica
    a[n_] := a[n] = Module[{ m, q}, {q, m} = QuotientRemainder[n, 4]; If[n < 4, If[n == 3, 1, 0], Sum[a[q + m - j]*(-1)^j*Binomial[m, j], {j, 0, m}]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 09 2018, from Maple *)

Formula

a(4*n) = a(n),
a(4*n+1) = a(n+1) - a(n),
a(4*n+2) = a(n+2) - 2*a(n+1) + a(n),
a(4*n+3) = a(n+3) - 3*a(n+2) + 3*a(n+1) - a(n).

A229655 Quintisection a(5n+k) gives k-th differences of a for k=0..4 with a(n)=0 for n<4 and a(4)=1.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 1, -4, 0, 0, 1, -3, 6, 0, 1, -2, 3, -4, 1, -1, 1, -1, 2, 0, 0, 0, 1, -8, 0, 0, 1, -7, 22, 0, 1, -6, 15, -28, 1, -5, 9, -13, 18, -4, 4, -4, 5, -11, 0, 0, 1, -6, 24, 0, 1, -5, 18, -46, 1, -4, 13, -28, 50, -3, 9, -15, 22, -33, 6, -6, 7
Offset: 0

Views

Author

Alois P. Heinz, Sep 27 2013

Keywords

Links

Crossrefs

Cf. A005590, A229653, A229654, A229656, A229657, A229658, A229659, A229660.

Programs

  • Maple
    a:= proc(n) option remember; local m, q;
      m:= irem(n, 5, 'q'); `if`(n<5, `if`(n=4, 1, 0),
      add(a(q+m-j)*(-1)^j*binomial(m, j), j=0..m))
    end:
seq(a(n), n=0..100);
  • Mathematica
    a[n_] := a[n] = Module[{ m, q}, {q, m} = QuotientRemainder[n, 5]; If[n < 5, If[n == 4, 1, 0], Sum[a[q + m - j]*(-1)^j*Binomial[m, j], {j, 0, m}]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 09 2018, from Maple *)

Formula

a(5*n) = a(n),
a(5*n+1) = a(n+1) - a(n),
a(5*n+2) = a(n+2) - 2*a(n+1) + a(n),
a(5*n+3) = a(n+3) - 3*a(n+2) + 3*a(n+1) - a(n),
a(5*n+4) = a(n+4) - 4*a(n+3) + 6*a(n+2) - 4*a(n+1) + a(n).

A229657 7-section a(7n+k) gives k-th differences of a for k=0..6 with a(n)=0 for n<6 and a(6)=1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, -6, 0, 0, 0, 0, 1, -5, 15, 0, 0, 0, 1, -4, 10, -20, 0, 0, 1, -3, 6, -10, 15, 0, 1, -2, 3, -4, 5, -6, 1, -1, 1, -1, 1, -1, 2, 0, 0, 0, 0, 0, 1, -12, 0, 0, 0, 0, 1, -11, 51, 0, 0, 0, 1, -10, 40, -110, 0, 0, 1, -9, 30, -70
Offset: 0

Views

Author

Alois P. Heinz, Sep 27 2013

Keywords

Links

Crossrefs

Cf. A005590, A229653, A229654, A229655, A229656, A229658, A229659, A229660.

Programs

  • Maple
    a:= proc(n) option remember; local m, q;
      m:= irem(n, 7, 'q'); `if`(n<7, `if`(n=6, 1, 0),
      add(a(q+m-j)*(-1)^j*binomial(m, j), j=0..m))
    end:
seq(a(n), n=0..100);

Formula

a(7*n+k) = Sum_{j=0..k} (-1)^j * C(k,j) * a(n+k-j) for k=0..6.

A229658 8-section a(8n+k) gives k-th differences of a for k=0..7 with a(n)=0 for n<7 and a(7)=1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, -7, 0, 0, 0, 0, 0, 1, -6, 21, 0, 0, 0, 0, 1, -5, 15, -35, 0, 0, 0, 1, -4, 10, -20, 35, 0, 0, 1, -3, 6, -10, 15, -21, 0, 1, -2, 3, -4, 5, -6, 7, 1, -1, 1, -1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 1, -14, 0, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Alois P. Heinz, Sep 27 2013

Keywords

Links

Crossrefs

Cf. A005590, A229653, A229654, A229655, A229656, A229657, A229659, A229660.

Programs

  • Maple
    a:= proc(n) option remember; local m, q;
      m:= irem(n, 8, 'q'); `if`(n<8, `if`(n=7, 1, 0),
      add(a(q+m-j)*(-1)^j*binomial(m, j), j=0..m))
    end:
seq(a(n), n=0..100);

Formula

a(8*n+k) = Sum_{j=0..k} (-1)^j * C(k,j) * a(n+k-j) for k=0..7.

A229659 9-section a(9n+k) gives k-th differences of a for k=0..8 with a(n)=0 for n<8 and a(8)=1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, -8, 0, 0, 0, 0, 0, 0, 1, -7, 28, 0, 0, 0, 0, 0, 1, -6, 21, -56, 0, 0, 0, 0, 1, -5, 15, -35, 70, 0, 0, 0, 1, -4, 10, -20, 35, -56, 0, 0, 1, -3, 6, -10, 15, -21, 28, 0, 1, -2, 3, -4, 5, -6, 7, -8, 1, -1, 1, -1
Offset: 0

Views

Author

Alois P. Heinz, Sep 27 2013

Keywords

Links

Crossrefs

Cf. A005590, A229653, A229654, A229655, A229656, A229657, A229658, A229660.

Programs

  • Maple
    a:= proc(n) option remember; local m, q;
      m:= irem(n, 9, 'q'); `if`(n<9, `if`(n=8, 1, 0),
      add(a(q+m-j)*(-1)^j*binomial(m, j), j=0..m))
    end:
seq(a(n), n=0..100);

Formula

a(9*n+k) = Sum_{j=0..k} (-1)^j * C(k,j) * a(n+k-j) for k=0..8.

A229660 10-section a(10n+k) gives k-th differences of a for k=0..9 with a(n)=0 for n<9 and a(9)=1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -9, 0, 0, 0, 0, 0, 0, 0, 1, -8, 36, 0, 0, 0, 0, 0, 0, 1, -7, 28, -84, 0, 0, 0, 0, 0, 1, -6, 21, -56, 126, 0, 0, 0, 0, 1, -5, 15, -35, 70, -126, 0, 0, 0, 1, -4, 10, -20, 35, -56, 84, 0, 0, 1, -3, 6, -10
Offset: 0

Views

Author

Alois P. Heinz, Sep 27 2013

Keywords

Links

Crossrefs

Cf. A005590, A229653, A229654, A229655, A229656, A229657, A229658, A229659.

Programs

  • Maple
    a:= proc(n) option remember; local m, q;
      m:= irem(n, 10, 'q'); `if`(n<10, `if`(n=9, 1, 0),
      add(a(q+m-j)*(-1)^j*binomial(m, j), j=0..m))
    end:
seq(a(n), n=0..100);

Formula

a(10*n+k) = Sum_{j=0..k} (-1)^j * C(k,j) * a(n+k-j) for k=0..9.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.