A110551 -id:A110551 - cp's OEIS frontend

A271832 Period 12 zigzag sequence: repeat [0,1,2,3,4,5,6,5,4,3,2,1].

0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1

Offset: 0

Wesley Ivan Hurt, Apr 15 2016

a(n)/36 is the probability that the sum shown after rolling a pair of standard dice is 1+(n mod 12). - Mathew Englander, Jul 11 2022

Decimal expansion of 37037/3000003. - Elmo R. Oliveira, Mar 03 2024

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

Period k zigzag sequences: A000035 (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), this sequence (k=12), A279313 (k=14), A279319 (k=16), A158289 (k=18).

Cf. A010677, A011655, A110551.

&cat[[0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1]: n in [0..10]];

A271832:=n->[0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1][(n mod 12)+1]: seq(A271832(n), n=0..300);

CoefficientList[Series[x*(1 + x + x^2 + x^3 + x^4 + x^5)/(1 - x + x^6 - x^7), {x, 0, 100}], x]

lista(nn) = for(n=0, nn, print1(abs(n-12*round(n/12)), ", ")); \\ Altug Alkan, Apr 15 2016

G.f.: x*(1 + x + x^2 + x^3 + x^4 + x^5)/(1 - x + x^6 - x^7).
a(n) = a(n-1) - a(n-6) + a(n-7) for n>6.
a(n) = abs(n - 12*round(n/12)).
a(n) = Sum_{i=1..n} (-1)^floor((i-1)/6).
a(2n) = a(10n) = 2*A260686(n), a(2n+1) = A110551(n).
a(3n) = 3*A007877(n), a(4n) = a(8n) = 4*A011655(n).
a(6n) = A010677(n) = 6*A000035(n).
a(n) = a(n-12) for n >= 12. - Wesley Ivan Hurt, Sep 07 2022

A110569 Period 6: repeat [2, 1, 3, 3, 1, 2].

Original entry on oeis.org

2, 1, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2, 2, 1, 3, 3, 1, 2

Offset: 0

Paul Barry, Jul 27 2005

Permutation of {1, 2, 3}, followed by its reversal, repeated.

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

Cf. A078008, A105198, A110549, A110550, A110551, A110568.

&cat [[2, 1, 3, 3, 1, 2]^^30]; // Wesley Ivan Hurt, Jun 27 2016

A110569:=n->[2, 1, 3, 3, 1, 2][(n mod 6)+1]: seq(A110569(n), n=0..100); # Wesley Ivan Hurt, Jun 27 2016

PadRight[{}, 100, {2, 1, 3, 3, 1, 2}] (* Wesley Ivan Hurt, Jun 27 2016 *)

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

a(n) = 1+(A078008(n) mod 3).
G.f.: (2-x+4*x^2-x^3+2*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) = 2 + 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 27 2016

Name changed by Wesley Ivan Hurt, Jun 27 2016

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

A206543 Period 10: repeat 1, 3, 5, 7, 9, 9, 7, 5, 3, 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 9, 7, 5, 3, 1, 1

Offset: 1

Wolfdieter Lang, Feb 09 2012

For general Modd n (not to be confused with mod n) see a comment on A203571. The present sequence gives the residues Modd 11 for the positive odd numbers not divisible by 11, which are given in A204454.

The underlying period length 22 sequence with offset 0 is P_11, also called Modd11, periodic([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]).

			Residue Modd 11 of the positive odd numbers not divisible by 11:
A204454: 1, 3, 5, 7, 9, 13, 15, 17, 19, 21, 23, 25, 27, ...
Modd 11: 1, 3, 5, 7, 9,  9,  7,  5,  3,  1,  1,  3,  5, ...

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,1).

Cf. A000012 (Modd 3), A084101 (Modd 5), A110551 (Modd 7).

PadRight[{},120,{1,3,5,7,9,9,7,5,3,1}] (* or *) LinearRecurrence[{2,-2,2,-2,1},{1,3,5,7,9},120] (* Harvey P. Dale, Oct 15 2017 *)

a(n)=[1, 3, 5, 7, 9, 9, 7, 5, 3, 1][n%10+1] \\ Charles R Greathouse IV, Jul 17 2016

a(n) = A204454(n) (Modd 11) := Modd11(A204454(n)), with the periodic sequence Modd11 with period length 22 given in the comment section.

O.g.f.: x*(1+x^9+3*x*(1+x^7)+5*x^2*(1+x^5)+7*x^3*(1+x^3)+9*x^4*(1+x))/(1-x^10) = x*(1+x)*(1-x^5)/((1+x^5)*(1-x)^2).

A206544 Period 12: repeat 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1

Offset: 1

Wolfdieter Lang, Feb 09 2012

For general Modd n (not to be confused with mod n) see a comment on A203571. The present sequence gives the residues Modd 13 of the positive odd numbers not divisible by 13, which are given in A204457.

The underlying periodic sequence with period length 26 is periodic([0,1,2,3,4,5,6,7,8,9,10,11,12,0,12,11,10,9,8,7,6,5,4,3,2,1]), called, with offset 0, P_13 or Modd13.

			Residue Modd 13 of the positive odd numbers not divisible by 13:
A204457: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, ...
Modd 13: 1, 3, 5, 7, 9, 11, 11,  9,  7,  5,  3,  1,  1,  3,  5,  7, ...

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

Cf. A000012 (Modd 3), A084101 (Modd 5), A110551 (Modd 7), A206543 (Modd 11).

LinearRecurrence[{1, 0, 0, 0, 0, -1, 1},{1, 3, 5, 7, 9, 11, 11},72] (* Ray Chandler, Aug 08 2015 *)

a(n)=[1, 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3][n%12+1] \\ Charles R Greathouse IV, Jul 17 2016

a(n) = A204457(n) (Modd 13) := Modd13(A204457(n)), n>=1, with the period length 26 periodic sequence Modd13 given in the comment section.

O.g.f.: x*(1+x^11+3*x*(1+x^9)+5*x^2*(1+x^7)+7*x^3*(1+x^5)+9*x^4*(1+x^3)+11*x^5*(1+x))/(1-x^12) = x*(1-x^6)*(1+x)/((1+x^6)*(1-x)^2).

A206545 Period length 16: repeat 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1, 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1

Offset: 1

Wolfdieter Lang, Feb 09 2012

For general Modd n see a comment on A203571. This sequence gives the Modd 17 residues of the odd numbers not divisible by 17, which are given in A204458.

The underlying periodic sequence with period length 34 is periodic (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 4, 3, 2, 1). This sequence with offset 0 is called P_17 or Modd17.

			Residue Modd 17 of the positive odd numbers not divisible by 17:
A204458: 1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29,...
Modd 17: 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11,  9,  7,  5,...

Cf. A000012 (Modd 3), A084101 (Modd 5), A110551 (Modd 7), A206543 (Modd 11), A206544 (Modd 13).

PadRight[{},120,Join[Range[1,15,2],Range[15,1,-2]]] (* Harvey P. Dale, Sep 21 2018 *)

a(n) = A204458(n) (Modd 17) := Modd17(A204458(n)), n>=1, with the periodic sequence Modd17, with period length 34, defined in the comment section.

O.g.f.: x*(1+x^15+3*x*(1+x^13)+5*x^2*(1+x^11)+7*x^3*(1+x^9)+9*x^4*(1+x^7)+11*x^5*(1+x^5)+ 13*x^6*(1+x^3)+15*x^7*(1+x))/(1-x^16) = x*(1+x)^2*(1+x^2)*(1+x^4)/((1+x^8)*(1-x)).

A234255 Decimal expansion of -B(12) = 691/2730, 13th Bernoulli number without sign.

Original entry on oeis.org

0, 2, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5, 3, 1, 1, 3, 5, 5

Offset: 1

Paul Curtz, Dec 22 2013

Essentially of period 6: repeat [5, 3, 1, 1, 3, 5] = A110551(n+3).
691*3663 = 2531133. See A021277.
Seventh part of the constant c=0.6323809537553113569215686274509803711... .
B(24) - B(12) = -86580. See A002882.

			0.2531135531135531135531135531135531135531135...

Index entries for linear recurrences with constant coefficients, signature (2,-2,1).

Cf. (A027641 or A164555)/A027642, A000367/A002445, A020793, A021046, A234355, A234356, A112828. B(24).

Cf. A002882, A110551, A021277.

[0,2] cat &cat [[5, 3, 1, 1, 3, 5]^^30]; // Wesley Ivan Hurt, Jun 28 2016

A234255:=n->[5, 3, 1, 1, 3, 5][(n mod 6)+1]: 0,2,seq(A234255(n), n=0..100); # Wesley Ivan Hurt, Jun 28 2016

Join[{0},RealDigits[-BernoulliB[12],10,120][[1]]] (* or *) PadRight[{0,2}, 120, {3,5,5,3,1,1}] (* Harvey P. Dale, Dec 30 2013 *)

default(realprecision, 120);
-bernfrac(12) + 0. \\ Rick L. Shepherd, Jan 15 2014

From Chai Wah Wu, Jun 04 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + a(n-3) for n > 5.
G.f.: x^2*(2 + x - 3*x^2 + 3*x^3)/((1 - x)*(1 - x + x^2)). (End)
From Wesley Ivan Hurt, Jun 28 2016: (Start)
a(n) = a(n-6) for n>8.
a(n) = (9 - 6*cos(n*Pi/3) + 2*sqrt(3)*sin(n*Pi/3))/3 for n>2. (End)

Offset corrected by and more terms from Rick L. Shepherd, Jan 15 2014

cp's OEIS Frontend

A271832 Period 12 zigzag sequence: repeat [0,1,2,3,4,5,6,5,4,3,2,1].

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A110569 Period 6: repeat [2, 1, 3, 3, 1, 2].

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

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A206543 Period 10: repeat 1, 3, 5, 7, 9, 9, 7, 5, 3, 1.

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A206544 Period 12: repeat 1, 3, 5, 7, 9, 11, 11, 9, 7, 5, 3, 1.

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A206545 Period length 16: repeat 1, 3, 5, 7, 9, 11, 13, 15, 15, 13, 11, 9, 7, 5, 3, 1.

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A234255 Decimal expansion of -B(12) = 691/2730, 13th Bernoulli number without sign.

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