A157742 -id:A157742 - cp's OEIS frontend

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

Offset: 0

Paul Curtz, Mar 12 2009

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.

Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1).

Cf. A157742, A158012.

&cat[[1, 2, 2, 1, 5, 5]: n in [0..20]]; // Wesley Ivan Hurt, Jun 17 2016

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

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 *)

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

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

Paul Curtz, Mar 12 2009

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

Also the decimal expansion of 6733370/111111111.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

Cf. A157742, A158012, A158068.

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)).

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

R. J. Mathar, Jan 28 2008

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

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 1).

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 *)

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

Paul Curtz, Mar 14 2009

nonn

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.

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

Edited and extended by R. J. Mathar, Apr 09 2009

cp's OEIS Frontend

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

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

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

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

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