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 1-8 of 8 results.

A229817 Even bisection gives sequence a itself, n->a(2*(2*n+k)-1) gives k-th differences of a for k=1..2 with a(n)=n for n<2.

Original entry on oeis.org

0, 1, 1, -1, 1, 0, -1, -2, 1, -2, 0, 4, -1, 2, -2, -3, 1, -1, -2, 0, 0, -1, 4, 0, -1, -1, 2, 4, -2, 3, -3, -6, 1, -3, -1, 5, -2, 2, 0, 2, 0, 4, -1, -9, 4, -5, 0, 8, -1, 3, -1, -7, 2, -4, 4, 3, -2, -1, 3, 5, -3, 4, -6, -6, 1, -2, -3, 1, -1, -1, 5, 3, -2, 2, 2
Offset: 0

Views

Author

Alois P. Heinz, Sep 30 2013

Keywords

Links

Crossrefs

Cf. A005590, A229818, A229819, A229820, A229821, A229822, A229823, A229824, A229825.

Programs

  • Maple
    a:= proc(n) option remember; local m, q, r;
      m:= (irem(n, 4, '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}, {q, m} = QuotientRemainder[n, 4]; m = (m + 1)/2; If[n<2, n, If[Mod[n, 2]==0, a[Quotient[n, 2]], Sum[a[q+m-j] * (-1)^j * Binomial[m, j], {j, 0, m}]]]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 22 2017, translated from Maple *)

Formula

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

A229818 Even bisection gives sequence a itself, n->a(2*(3*n+k)-1) gives k-th differences of a for k=1..3 with a(n)=n for n<2.

Original entry on oeis.org

0, 1, 1, -1, 1, -1, -1, 0, 1, -2, -1, 6, -1, -2, 0, 4, 1, -8, -2, 2, -1, -4, 6, 6, -1, -2, -2, 2, 0, -1, 4, 0, 1, 1, -8, -1, -2, 1, 2, 0, -1, -4, -4, 1, 6, -4, 6, 8, -1, -3, -2, 4, -2, 2, 2, 1, 0, 6, -1, -20, 4, 7, 0, -14, 1, 20, 1, -7, -8, 6, -1, -3, -2, -1
Offset: 0

Views

Author

Alois P. Heinz, Sep 30 2013

Keywords

Links

Crossrefs

Cf. A005590, A229817, A229819, A229820, A229821, A229822, A229823, A229824, A229825.

Programs

  • Maple
    a:= proc(n) option remember; local m, q, r;
      m:= (irem(n, 6, '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, 6]; 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 *)
  • PARI
    {M=Map(); a(n)= n&&n>>=valuation(n, 2); my(r); mapisdefined(M, n, &r) && return(r); r=if(n<2, n, my(m=n%6, k=n\6); if(1==m, a(k+1)-a(k), 3==m, a(k+2)-2*a(k+1)+a(k), a(k+3)-3*a(k+2)+3*a(k+1)-a(k))); mapput(~M, n, r); r;} \\ Ruud H.G. van Tol, Nov 19 2024

Formula

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

A229819 Even bisection gives sequence a itself, n->a(2*(4*n+k)-1) gives k-th differences of a for k=1..4 with a(n)=n for n<2.

Original entry on oeis.org

0, 1, 1, -1, 1, -1, -1, 7, 1, 0, -1, -2, -1, 6, 7, -14, 1, -2, 0, 4, -1, -8, -2, 14, -1, 2, 6, -4, 7, 6, -14, 0, 1, -2, -2, 2, 0, 6, 4, -28, -1, 0, -8, 8, -2, -22, 14, 41, -1, 8, 2, -14, 6, 19, -4, -24, 7, -6, 6, 5, -14, -5, 0, 5, 1, -1, -2, 0, -2, 0, 2, 2, 0
Offset: 0

Views

Author

Alois P. Heinz, Sep 30 2013

Keywords

Links

Crossrefs

Cf. A005590, A229817, A229818, A229820, A229821, A229822, A229823, A229824, A229825.

Programs

  • Maple
    a:= proc(n) option remember; local m, q, r;
      m:= (irem(n, 8, '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, 8]; 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(8*n+1) = a(n+1) - a(n),
a(8*n+3) = a(n+2) - 2*a(n+1) + a(n),
a(8*n+5) = a(n+3) - 3*a(n+2) + 3*a(n+1) - a(n).
a(8*n+7) = a(n+4) - 4*a(n+3) + 6*a(n+2) - 4*a(n+1) + a(n).

A229820 Even bisection gives sequence a itself, n->a(2*(5*n+k)-1) gives k-th differences of a for k=1..5 with a(n)=n for n<2.

Original entry on oeis.org

0, 1, 1, -1, 1, -1, -1, 7, 1, -21, -1, 0, -1, -2, 7, 6, 1, -14, -21, 28, -1, -2, 0, 4, -1, -8, -2, 14, 7, -14, 6, 2, 1, -4, -14, 6, -21, 0, 28, -28, -1, -2, -2, 2, 0, 6, 4, -28, -1, 48, -8, 0, -2, 8, 14, -22, 7, 20, -14, 40, 6, 8, 2, -14, 1, -2, -4, 60, -14
Offset: 0

Views

Author

Alois P. Heinz, Sep 30 2013

Keywords

Links

Crossrefs

Cf. A005590, A229817, A229818, A229819, A229821, A229822, A229823, A229824, A229825.

Programs

  • Maple
    a:= proc(n) option remember; local m, q, r;
      m:= (irem(n, 10, '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, 10]; 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*(5*n+k)-1) = Sum_{j=0..k} (-1)^j * C(k,j) * a(n+k-j) for k=1..5.

A229822 Even bisection gives sequence a itself, n->a(2*(7*n+k)-1) gives k-th differences of a for k=1..7 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, 0, 1, -2, -21, 6, -1, -14, 49, 28, -1, -42, -91, 28, 7, -2, 0, 4, 1, -8, -2, 14, -21, -14, 6, -14, -1, 90, -14, 2, 49, -4, 28, 6, -1, 0, -42, -28, -91, 76, 28, -84, 7, -2, -2, 2, 0, 6, 4, -28, 1, 48, -8
Offset: 0

Views

Author

Alois P. Heinz, Sep 30 2013

Keywords

Links

Crossrefs

Cf. A005590, A229817, A229818, A229819, A229820, A229821, A229823, A229824, A229825.

Programs

  • Maple
    a:= proc(n) option remember; local m, q, r;
      m:= (irem(n, 14, '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, 14]; 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*(7*n+k)-1) = Sum_{j=0..k} (-1)^j * C(k,j) * a(n+k-j) for k=1..7.

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.
Showing 1-8 of 8 results.

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