A088689 -id:A088689 - cp's OEIS frontend

A088688 Binomial transform of A088689.

0, 1, 3, 6, 12, 27, 63, 141, 297, 594, 1146, 2169, 4095, 7827, 15291, 30582, 62256, 127791, 262143, 534129, 1078101, 2156202, 4282878, 8477181, 16777215, 33288711, 66311703, 132623406, 266043972, 534479427, 1073741823, 2154658101

Offset: 0

Paul Barry, Oct 06 2003

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6).

Cf. A001045.

Table[Sum[Binomial[n,k] * Mod[k*Floor[3*(k+1)/2] - 2*k, 3], {k, 0, n}], {n, 0, 40}] (* Vaclav Kotesovec, Oct 30 2017 *)
CoefficientList[Series[-x(1-3x+3x^2+x^3)/((2x-1)(x^2-x+1)(3x^2-3x+1)),{x,0,40}],x] (* or *) LinearRecurrence[{6,-15,20,-15,6},{0,1,3,6,12},40] (* Harvey P. Dale, Aug 28 2025 *)

a(n) = 2^n - cos(Pi*n/3) - 3^(n/2)*sin(Pi*n/6)/sqrt(3).

O.g.f.: -x(1-3x+3x^2+x^3)/[(2x-1)(x^2-x+1)(3x^2-3x+1)]. - R. J. Mathar, Apr 02 2008

A175286 Pisano period of the Jacobsthal sequence A001045 modulo n.

Original entry on oeis.org

1, 1, 6, 2, 4, 6, 6, 2, 18, 4, 10, 6, 12, 6, 12, 2, 8, 18, 18, 4, 6, 10, 22, 6, 20, 12, 54, 6, 28, 12, 10, 2, 30, 8, 12, 18, 36, 18, 12, 4, 20, 6, 14, 10, 36, 22, 46, 6, 42, 20, 24, 12, 52, 54, 20, 6, 18, 28, 58, 12, 60, 10, 18, 2, 12, 30, 66, 8, 66, 12, 70, 18, 18, 36, 60, 18, 30, 12, 78, 4

Offset: 1

R. J. Mathar, Mar 21 2010

nonn

			Reading the sequence 0, 1, 1, 3, 5, 11, 21, ... modulo n=3, we get 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, ... = A088689, which has a period (1, 1, 0, 2, 2, 0) of length a(n=3) = 6.

Eric Weisstein's World of Mathematics, Pisano period.
Wikipedia, Pisano period.

Cf. A001045, A001175, A088689, A175181.

A110568 Period 6: repeat [1, 0, 2, 2, 0, 1].

Original entry on oeis.org

1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2

Offset: 0

Paul Barry, Jul 27 2005

Permutation of {0, 1, 2}, followed by its reversal, repeated.

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

Cf. A078008, A088689, A105198, A110549, A110550, A110551, A110569.

&cat [[1, 0, 2, 2, 0, 1]^^30]; // Wesley Ivan Hurt, Jun 28 2016

A110568:=n->[1, 0, 2, 2, 0, 1][(n mod 6)+1]: seq(A110568(n), n=0..100); # Wesley Ivan Hurt, Jun 28 2016

Mod[#,3]&/@CoefficientList[Series[(1-x)/(1-x-2x^2),{x,0,100}],x] (* Harvey P. Dale, Mar 30 2011 *)
PadRight[{}, 100, {1, 0, 2, 2, 0, 1}] (* Wesley Ivan Hurt, Jun 28 2016 *)
LinearRecurrence[{1,-1,1,-1,1},{1,0,2,2,0},100] (* Harvey P. Dale, Apr 03 2019 *)

x='x+O('x^50); Vec((1-x+3*x^2-x^3+x^4)/(1-x+x^2-x^3+x^4-x^5)) \\ G. C. Greubel, Aug 31 2017

a(n) = A078008(n) mod 3.
G.f.: (1-x+3*x^2-x^3+x^4) / (1-x+x^2-x^3+x^4-x^5).
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = 1 + cos(2*Pi*n/3)/2 - sqrt(3)*sin(2*Pi*n/3)/2 - cos(Pi*n/3)/2 + sqrt(3)*sin(Pi*n/3)/6.
a(n) = a(n-6) for n > 5. - Wesley Ivan Hurt, Jun 28 2016
a(n) = ((n-1)*(-1)^(n-1) mod 3). - Wesley Ivan Hurt, Jan 07 2021

Name changed by Wesley Ivan Hurt, Jun 28 2016

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A088688 Binomial transform of A088689.

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A175286 Pisano period of the Jacobsthal sequence A001045 modulo n.

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A110568 Period 6: repeat [1, 0, 2, 2, 0, 1].

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