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.

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A229823 Even bisection gives sequence a itself, n->a(2*(8*n+k)-1) gives k-th differences of a for k=1..8 with a(n)=n for n<2.

Original entry on oeis.org

0, 1, 1, -1, 1, -1, -1, 7, 1, -21, -1, 49, -1, -91, 7, 119, 1, 0, -21, -2, -1, 6, 49, -14, -1, 28, -91, -42, 7, 28, 119, 62, 1, -2, 0, 4, -21, -8, -2, 14, -1, -14, 6, -14, 49, 90, -14, -174, -1, 2, 28, -4, -91, 6, -42, 0, 7, -28, 28, 76, 119, -84, 62, -78, 1
Offset: 0

Views

Author

Alois P. Heinz, Sep 30 2013

Keywords

Links

Crossrefs

Cf. A005590, A229817, A229818, A229819, A229820, A229821, A229822, A229824, A229825.

Programs

  • Maple
    a:= proc(n) option remember; local m, q, r;
      m:= (irem(n, 16, 'q')+1)/2;
      `if`(n<2, n, `if`(irem(n, 2, 'r')=0, a(r),
      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, r, q2, r2}, {q, r} = QuotientRemainder[n, 16]; m = (r+1)/2; If[n<2, n, {q2, r2} = QuotientRemainder[n, 2]; If[r2 == 0, a[q2], Sum[a[q+m-j]*(-1)^j*Binomial[m, j], {j, 0, m}]]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 08 2017, translated from Maple *)

Formula

a(2*n) = a(n),
a(2*(8*n+k)-1) = Sum_{j=0..k} (-1)^j * C(k,j) * a(n+k-j) for k=1..8.

A229824 Even bisection gives sequence a itself, n->a(2*(9*n+k)-1) gives k-th differences of a for k=1..9 with a(n)=n for n<2.

Original entry on oeis.org

0, 1, 1, -1, 1, -1, -1, 7, 1, -21, -1, 49, -1, -91, 7, 119, 1, -57, -21, 0, -1, -2, 49, 6, -1, -14, -91, 28, 7, -42, 119, 28, 1, 62, -57, -236, -21, -2, 0, 4, -1, -8, -2, 14, 49, -14, 6, -14, -1, 90, -14, -174, -91, 96, 28, 2, 7, -4, -42, 6, 119, 0, 28, -28, 1
Offset: 0

Views

Author

Alois P. Heinz, Sep 30 2013

Keywords

Links

Crossrefs

Cf. A005590, A229817, A229818, A229819, A229820, A229821, A229822, A229823, A229825.

Programs

  • Maple
    a:= proc(n) option remember; local m, q, r;
      m:= (irem(n, 18, 'q')+1)/2;
      `if`(n<2, n, `if`(irem(n, 2, 'r')=0, a(r),
      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, r, q2, r2}, {q, r} = QuotientRemainder[n, 18]; m = (r+1)/2; If[n<2, n, {q2, r2} = QuotientRemainder[n, 2]; If[r2 == 0, a[q2], Sum[a[q+m-j]*(-1)^j*Binomial[m, j], {j, 0, m}]]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 08 2017, translated from Maple *)

Formula

a(2*n) = a(n),
a(2*(9*n+k)-1) = Sum_{j=0..k} (-1)^j * C(k,j) * a(n+k-j) for k=1..9.

A229825 Even bisection gives sequence a itself, n->a(2*(10*n+k)-1) gives k-th differences of a for k=1..10 with a(n)=n for n<2.

Original entry on oeis.org

0, 1, 1, -1, 1, -1, -1, 7, 1, -21, -1, 49, -1, -91, 7, 119, 1, -57, -21, -179, -1, 0, 49, -2, -1, 6, -91, -14, 7, 28, 119, -42, 1, 28, -57, 62, -21, -236, -179, 332, -1, -2, 0, 4, 49, -8, -2, 14, -1, -14, 6, -14, -91, 90, -14, -174, 7, 96, 28, 396, 119, 2, -42
Offset: 0

Views

Author

Alois P. Heinz, Sep 30 2013

Keywords

Links

Crossrefs

Cf. A005590, A229817, A229818, A229819, A229820, A229821, A229822, A229823, A229824.

Programs

  • Maple
    a:= proc(n) option remember; local m, q, r;
      m:= (irem(n, 20, 'q')+1)/2;
      `if`(n<2, n, `if`(irem(n, 2, 'r')=0, a(r),
      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, r, q2, r2}, {q, r} = QuotientRemainder[n, 20]; m = (r+1)/2; If[n<2, n, {q2, r2} = QuotientRemainder[n, 2]; If[r2 == 0, a[q2], Sum[a[q+m-j]*(-1)^j*Binomial[m, j], {j, 0, m}]]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 08 2017, translated from Maple *)

Formula

a(2*n) = a(n),
a(2*(10*n+k)-1) = Sum_{j=0..k} (-1)^j * C(k,j) * a(n+k-j) for k=1..10.

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).

A229656 6-section a(6n+k) gives k-th differences of a for k=0..5 with a(n)=0 for n<5 and a(5)=1.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, -5, 0, 0, 0, 1, -4, 10, 0, 0, 1, -3, 6, -10, 0, 1, -2, 3, -4, 5, 1, -1, 1, -1, 1, 0, 0, 0, 0, 0, 1, -10, 0, 0, 0, 1, -9, 35, 0, 0, 1, -8, 26, -60, 0, 1, -7, 18, -34, 55, 1, -6, 11, -16, 21, -25, -5, 5, -5, 5, -4, -4, 0, 0, 0, 1
Offset: 0

Views

Author

Alois P. Heinz, Sep 27 2013

Keywords

Links

Crossrefs

Cf. A005590, A229653, A229654, A229655, A229657, A229658, A229659, A229660.

Programs

  • Maple
    a:= proc(n) option remember; local m, q;
      m:= irem(n, 6, 'q'); `if`(n<6, `if`(n=5, 1, 0),
      add(a(q+m-j)*(-1)^j*binomial(m, j), j=0..m))
    end:
seq(a(n), n=0..100);

Formula

a(6*n+k) = Sum_{j=0..k} (-1)^j * C(k,j) * a(n+k-j) for k=0..5.

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.
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