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 1-3 of 3 results.

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

Links

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

A158137 Period 9: repeat [-2,4,-2,4,-2,-2,1,1,-2].

Original entry on oeis.org

-2, 4, -2, 4, -2, -2, 1, 1, -2, -2, 4, -2, 4, -2, -2, 1, 1, -2, -2, 4, -2, 4, -2, -2, 1, 1, -2, -2, 4, -2, 4, -2, -2, 1, 1, -2, -2, 4, -2, 4, -2, -2, 1, 1, -2, -2, 4, -2, 4, -2, -2, 1, 1, -2, -2, 4, -2, 4, -2, -2, 1, 1, -2, -2, 4, -2, 4, -2, -2, 1, 1, -2, -2, 4, -2, 4, -2, -2, 1, 1, -2, -2, 4, -2
Offset: 0

Views

Author

Paul Curtz, Mar 13 2009

Keywords

Comments

Rotated by 7 places, the period is palindromic: 1,-2,-2,4,-2,4,-2,-2,1.

Links

Crossrefs

Cf. A156346.

Programs

Formula

a(n)= A158090(n)-2 = A158090(n)-A007395(n+1).
G.f.: (-2+2*x+4*x^3+x^6+2*x^4+2*x^7)/((1+x+x^2)*(x^6+x^3+1)). a(n)= -a(n-1)-a(n-2)-a(n-3)-a(n-4)-a(n-5)-a(n-6)-a(n-7)-a(n-8). - R. J. Mathar, Apr 09 2009

Extensions

Edited and extended by R. J. Mathar, Apr 09 2009
Showing 1-3 of 3 results.

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