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 1-4 of 4 results.

A158068 Period 6: repeat [1, 2, 2, 1, 5, 5].

Original entry on oeis.org

1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2, 1, 5, 5, 1, 2, 2
Offset: 0

Views

Author

Paul Curtz, Mar 12 2009

Keywords

Comments

The sequence can be generated starting an array T(n,k) by placing the
periodic sequence 1,2,5 (repeat 1,2,5) in the top row n=0, then defining
the next rows by T(n+1,k) = T(n,k)*T(n,k+1) mod 9, which all have a period T(n,k)=T(n,k+3).
One finds the periodicity T(n+6,k)=T(n,k), and then defines a(n)=T(n,1).
Also the partial fraction expansion of (85+sqrt(12469))/138.
Also the decimal expansion of 11105/90909.

Crossrefs

Programs

  • Magma
    &cat[[1, 2, 2, 1, 5, 5]: n in [0..20]]; // Wesley Ivan Hurt, Jun 17 2016
  • Maple
    A158068:=n->(8-5*cos(2*n*Pi/3)-3*sqrt(3)*sin(n*Pi/3))/3: seq(A158068(n), n=0..100); # Wesley Ivan Hurt, Jun 17 2016
  • Mathematica
    PadRight[{}, 120, {1,2,2,1,5,5}] (* or *) LinearRecurrence[{1,-1,1,-1,1},{1,2,2,1,5}, 120] (* Harvey P. Dale, Jan 27 2015 *)

Formula

a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) for n>4.
G.f.: (1+x+5*x^4+x^2)/((1-x)*(1-x+x^2)*(1+x+x^2)). [Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009]
a(n) = (8-5*cos(2*n*Pi/3)-3*sqrt(3)*sin(n*Pi/3))/3. - Wesley Ivan Hurt, Jun 17 2016

Extensions

Offset set to 0 - R. J. Mathar, Sep 17 2009

A158090 Period 9: repeat [0, 6, 0, 6, 0, 0, 3, 3, 0].

Original entry on oeis.org

0, 6, 0, 6, 0, 0, 3, 3, 0, 0, 6, 0, 6, 0, 0, 3, 3, 0, 0, 6, 0, 6, 0, 0, 3, 3, 0, 0, 6, 0, 6, 0, 0, 3, 3, 0, 0, 6, 0, 6, 0, 0, 3, 3, 0, 0, 6, 0, 6, 0, 0, 3, 3, 0, 0, 6, 0, 6, 0, 0, 3, 3, 0, 0, 6, 0, 6, 0, 0, 3, 3, 0, 0, 6, 0, 6, 0, 0, 3, 3, 0, 0, 6, 0, 6, 0, 0, 3, 3, 0, 0, 6, 0, 6, 0, 0, 3, 3, 0, 0, 6, 0, 6, 0, 0
Offset: 0

Views

Author

Paul Curtz, Mar 12 2009

Keywords

Comments

Also the continued fraction expansion of 6+sqrt(3970)/10 (dropping a(0)).
Also the decimal expansion of 6733370/111111111.

Crossrefs

Formula

a(n) = ( A061037(n)*A061037(n+1) ) mod 9.
a(n) = a(n-9). G.f.: -3*x*(2+2*x^2+x^5+x^6)/((x-1)*(1+x+x^2)*(x^6+x^3+1)).

Extensions

A-number in the formula corrected by R. J. Mathar, Sep 11 2009

A134804 Remainder of triangular number A000217(n) modulo 9.

Original entry on oeis.org

0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6, 3, 1, 0, 0, 1, 3, 6, 1, 6
Offset: 0

Views

Author

R. J. Mathar, Jan 28 2008

Keywords

Comments

Periodic with period 9 since A000217(n+9) = A000217(n)+9(n+5) .
From Jacobsthal numbers A001045, A156060 = 0,1,1,3,5,2,3,7,4,0,8, = b(n). a(n)=A156060(n)*A156060(n+1) mod 9. Same transform (a(n)*a(n+1) mod 9 or b(n)*b(n+1) mod 9) in A157742, A158012, A158068, A158090. - Paul Curtz, Mar 25 2009

Programs

  • Mathematica
    LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1},{0, 1, 3, 6, 1, 6, 3, 1, 0},105] (* Ray Chandler, Aug 26 2015 *)

Formula

a(n) = A010878(A000217(n)) = A010878(A055263(n)) = a(n-9).
O.g.f.: (-2x+2)/[3(x^2+x+1)]+(-3+3x^5)/(x^6+x^3+1)-7/[3(x-1)].

A158233 a(n) = A120070(n+1)*A120070(n+2) mod 9.

Original entry on oeis.org

6, 4, 3, 0, 3, 6, 0, 3, 0, 0, 4, 0, 0, 4, 6, 0, 0, 6, 0, 6, 0, 0, 6, 3, 0, 3, 6, 3, 4, 0, 0, 4, 0, 0, 4, 0, 0, 6, 3, 0, 3, 6, 0, 0, 3, 0, 0, 6, 0, 6, 0, 0, 3, 3, 6, 4, 0, 0, 4, 0, 0, 4, 0, 0, 4, 3, 0, 3, 0, 0, 0, 3, 0, 6, 6, 0, 3, 6, 0, 3, 0, 0, 0, 3, 0, 6, 6, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, 0, 4, 0, 0, 4, 6
Offset: 1

Views

Author

Paul Curtz, Mar 14 2009

Keywords

Comments

Conjecture: this contains only the numbers 0,3,4,6 (verified for the first 5000 terms).
This multiply-modulo transformation is also used in the unrelated A157742, A158012, A158068, A158090.

Programs

  • Maple
    A120070 := proc(m,n) if m-1 >= n then m^2-n^2; else 0; fi; end:
    A120070flat := proc(n) i := 2 ; for m from 2 do for l from 1 to m-1 do if i = n then RETURN(A120070(m,l)) ; else i := i+1 ; fi; od: od: end:
    A158233 := proc(n) (A120070flat(n+1)*A120070flat(n+2) ) mod 9 ; end: seq(A158233(n),n=1..180) ; # R. J. Mathar, Apr 09 2009

Extensions

Edited and extended by R. J. Mathar, Apr 09 2009
Showing 1-4 of 4 results.