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.

A077623 a(1)=1, a(2)=2, a(3)=4, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

1, 2, 4, 1, 5, 0, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10, 0, 11, 1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 6
Offset: 1

Views

Author

Benoit Cloitre, Dec 02 2002

Keywords

Links

Crossrefs

Cf. A077653, A079623, A079624, A080096, A088226.

Programs

  • Haskell
    a077623 n = a077623_list !! (n-1)
a077623_list = 1 : 2 : 4 : zipWith3 (\u v w -> abs (w - v - u))
               a077623_list (tail a077623_list) (drop 2 a077623_list)
-- Reinhard Zumkeller, Oct 11 2014
  • Magma
    m:=120;
A077623:=[n le 3 select 2^(n-1) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A077623[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024
  • Mathematica
    a[n_]:= a[n]= If[n<4, 2^(n-1), Abs[a[n-1] -a[n-2] -a[n-3]]];
Table[a[n], {n,120}] (* G. C. Greubel, Sep 11 2024 *)
  • SageMath
    @CachedFunction
def a(n): # a = A077623
    if n<4: return 2^(n-1)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,121)] # G. C. Greubel, Sep 11 2024

Formula

a(n)/sqrt(n) is bounded. More precisely, let M(n) = Max ( a(k) : 1<=k<=n ); then M(n) = floor(sqrt(n+29)) for n>=4

A080096 a(1) = a(2) = 1, a(3) = 2, thereafter a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

1, 1, 2, 0, 3, 1, 2, 2, 1, 3, 0, 4, 1, 3, 2, 2, 3, 1, 4, 0, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5
Offset: 1

Views

Author

Benoit Cloitre, Jan 28 2003

Keywords

Links

Crossrefs

Cf. A028347, A028557, A077623, A077653, A079623, A079624, A088226.

Programs

  • Haskell
    a080096 n = a080096_list !! (n-1)
a080096_list = 1 : 1 : 2 : zipWith3 (\u v w -> abs (w - v - u))
               a080096_list (tail a080096_list) (drop 2 a080096_list)
-- Reinhard Zumkeller, Oct 11 2014
  • Magma
    m:=120;
A080096:=[n le 3 select Floor((n+1)/2) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A080096[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024
  • Mathematica
    nxt[{a_,b_,c_}]:={b,c,Abs[c-b-a]}; NestList[nxt,{1,1,2},110][[All,1]] (* Harvey P. Dale, Nov 14 2021 *)
  • PARI
    a(n)=local(k,m); if(n<1,0,k=sqrtint(n+4); m=n+4-k^2; if(m%2,m\2+1,k-m\2))
  • SageMath
    @CachedFunction
def a(n): # a = A080096
    if n<4: return int((n+1)//2)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,101)] # G. C. Greubel, Sep 11 2024

Formula

For n>=3 Max( a(k) : 1<=k<=n ) = floor ( sqrt(n+4)).
Special cases: a(n^2 + 4*n - 1) = 0 and a(n^2 - 4) = n.
a(A028557(n)) = a(A028557(n+1)).
Sum_{k=(n-1)^2 .. n^2} a(k) = n^2.

A077653 a(1)=1, a(2)=2, a(3)=2, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

1, 2, 2, 1, 3, 0, 4, 1, 3, 2, 2, 3, 1, 4, 0, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 10, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7
Offset: 1

Views

Author

Benoit Cloitre, Dec 02 2002

Keywords

Comments

Conjecture : let z(1)=x; z(2)=y; z(3)= z; z(n)=abs(z(n-1)-z(n-2)-z(n-3)) if z(n) is unbounded (i.e. x,y,z are such that z(n) doesn't reach a cycle of length 2), then there are 2 integers n(x,y,z) and w(x,y,z) such that M(n) = floor(sqrt(n+w(x,y,z))) for n>n(,x,y,z) where M(n) = Max ( a(k) : 1<=k<=n ). As example : w(1,2,2)=9 n(1,2,2)=4; w(1,2,4)=29 n(1,2,4)=4; w(1,2,8)=157 n(1,2,8)=9

Links

Crossrefs

Cf. A077623, A079623, A079624, A080096, A088226.

Programs

  • Haskell
    a077653 n = a077653_list !! (n-1)
a077653_list = 1 : 2 : 2 : zipWith3 (\u v w -> abs (w - v - u))
               a077653_list (tail a077653_list) (drop 2 a077653_list)
-- Reinhard Zumkeller, Oct 11 2014
  • Magma
    m:=120;
A077653:=[n le 3 select Floor((n+2)/2) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A077653[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024
  • Mathematica
    nxt[{a_,b_,c_}]:={b,c,Abs[c-b-a]}; NestList[nxt,{1,2,2},110][[All,1]] (* Harvey P. Dale, Sep 01 2020 *)
  • SageMath
    @CachedFunction
def a(n): # a = A077653
    if n<4: return int((n+2)//2)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,101)] # G. C. Greubel, Sep 11 2024

Formula

a(n)/sqrt(n) is bounded. More precisely, let M(n) = Max ( a(k) : 1<=k<=n ); then M(n)= floor(sqrt(n+9)) for n>4

A079624 a(1) = a(2) = 1, a(3) = 6, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Jan 30 2003

Keywords

Links

Crossrefs

Cf. A077623, A077653, A078108, A079623, A080096, A088226.

Programs

  • Haskell
    a079624 n = a079624_list !! (n-1)
a079624_list = 1 : 1 : 6 : zipWith3 (\u v w -> abs (w - v - u))
               a079624_list (tail a079624_list) (drop 2 a079624_list)
-- Reinhard Zumkeller, Oct 11 2014
  • Magma
    m:=120;
A079624:=[n le 3 select 6^Floor((n-1)/2) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A079624[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024
  • Mathematica
    a[n_]:= a[n]= If[n<4, 6^Floor[(n-1)/2], Abs[a[n-1] -a[n-2] -a[n-3]]];
Table[a[n], {n,100}] (* G. C. Greubel, Sep 11 2024 *)
  • SageMath
    @CachedFunction
def a(n): # a = A079624
    if n<4: return 6^((n-1)//2)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,101)] # G. C. Greubel, Sep 11 2024

Formula

For n >= 5, a(n^2 + 24*n - 13) = 0.
For n >= 38, Max( a(k) : 1<=k<=n) = floor(sqrt(n+156)).

A088226 a(1)=0, a(2)=0, a(3)=1; for n>3, a(n) = abs(a(n-1) - a(n-2) - a(n-3)).

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 1, 1, 2, 0, 3, 1, 2, 2, 1, 3, 0, 4, 1, 3, 2, 2, 3, 1, 4, 0, 5, 1, 4, 2, 3, 3, 2, 4, 1, 5, 0, 6, 1, 5, 2, 4, 3, 3, 4, 2, 5, 1, 6, 0, 7, 1, 6, 2, 5, 3, 4, 4, 3, 5, 2, 6, 1, 7, 0, 8, 1, 7, 2, 6, 3, 5, 4, 4, 5, 3, 6, 2, 7, 1, 8, 0, 9, 1, 8, 2, 7, 3, 6, 4, 5, 5, 4, 6, 3, 7, 2, 8, 1, 9, 0, 10, 1, 9, 2
Offset: 1

Views

Author

Benoit Cloitre, Nov 04 2003

Keywords

Comments

Conjecture: a(n) = m/2 where m is the smallest even distance from n to a square. - Ralf Stephan, Sep 23 2013

Links

Crossrefs

Cf. A077623, A077653, A079623, A079624, A080096.

Programs

  • Haskell
    a088226 n = a088226_list !! (n-1)
a088226_list = 0 : 0 : 1 : zipWith3 (\u v w -> abs (w - v - u))
               a088226_list (tail a088226_list) (drop 2 a088226_list)
-- Reinhard Zumkeller, Oct 11 2014
  • Magma
    m:=120;
A088226:=[n le 3 select Floor((n-1)/2) else Abs(Self(n-1) - Self(n-2) - Self(n-3)): n in [1..m+5]];
[A088226[n]: n in [1..m]]; // G. C. Greubel, Sep 11 2024
  • Mathematica
    RecurrenceTable[{a[1]==a[2]==0,a[3]==1,a[n]==Abs[a[n-1]-a[n-2]-a[n-3]]},a,{n,110}] (* Harvey P. Dale, Apr 13 2012 *)
  • PARI
    a(n)=t=sqrtint(n);if((n-t*t)%2==0,(n-t*t)/2,((t+1)^2-n)/2) \\ Ralf Stephan, Sep 23 2013
  • SageMath
    @CachedFunction
def a(n): # a = A088226
    if n<4: return int((n-1)//2)
    else: return abs(a(n-1)-a(n-2)-a(n-3))
[a(n) for n in range(1,101)] # G. C. Greubel, Sep 11 2024

Formula

a(k^2 + 2*m + 2) = k-m and a(k^2 + 2*m + 1) = m, for k >= 0 and 0 <= m <= k.
Showing 1-5 of 5 results.

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