A110549 -id:A110549 - cp's OEIS frontend

A105198 a(n) = n(n+1)/2 mod 4.

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

Offset: 0

Oscar Takeshita, Apr 11 2005

0,1,3,2,2,3,1,0 repeated indefinitely.

If N is any power of 2 then n(n+1)/2 mod N is a repeating pattern of length 2N. Moreover, the first N digits form a permutation P of A={0,1,...,N-1}. The subsequent N digits are P in the reversed order. The technique is useful for the generation of arbitrarily large pseudo-random permutations.

Antti Karttunen, Table of n, a(n) for n = 0..8191
O. Y. Takeshita and D. J. Costello, Jr., New Deterministic Interleaver Designs for Turbo-Codes, IEEE Trans. Inform. Theory, vol. 46, no. 6, pp. 1988-2006, Sept. 2000.
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1).

Cf. triangular numbers A000217, A105332-A105340.

One less than A110549, A133882 shifted once right, with zero inserted to front.

[ -Ceiling(n/2)*(-1)^n mod 4 : n in [0..100]]; // Wesley Ivan Hurt, Jul 13 2014

for n from 0 to 300 do printf(`%d,`, n*(n+1)/2 mod 4) od: # James Sellers, Apr 21 2005
A105198:=n->-ceil(n/2)*(-1)^n mod 4: seq(A105198(n), n=0..100); # Wesley Ivan Hurt, Jul 13 2014

Table[Mod[-Ceiling[n/2] (-1)^n, 4], {n, 0, 100}] (* Wesley Ivan Hurt, Jul 13 2014 *)

Vec((x+2*x^2+2*x^4+x^5)/(1-x+x^2-x^3+x^4-x^5+x^6-x^7) + O(x^90)) \\ Michel Marcus, Jul 13 2014

def A105198(n): return (n*(n+1)>>1)&3 # Chai Wah Wu, Apr 17 2025

(define (A105198 n) (modulo (* 1/2 n (+ 1 n)) 4)) ;; Antti Karttunen, Aug 10 2017

From Paul Barry, Jul 26 2005: (Start)
G.f.: (x + 2x^2 + 2x^4 + x^5)/(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7).
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) = cos(3*Pi*n/4 + Pi/4)/2 + (1/2 - sqrt(2)/2)*sin(3*Pi*n/4 + Pi/4) - (1/2 + sqrt(2)/2)*cos(Pi*n/4 + Pi/4) - sin(Pi*n/4 + Pi/4)/2 - cos(Pi*n/2)/2 + sin(Pi*n/2)/2 + 3/2. (End)
a(n) = (((n+1)^5 - n^5 - 1) mod 120)/30. - Gary Detlefs, Mar 25 2012
a(n) = -ceiling(n/2)*(-1)^n mod 4. - Wesley Ivan Hurt, Jul 13 2014

More terms from James Sellers, Apr 21 2005

A110551 Period 6: repeat [1, 3, 5, 5, 3, 1].

Original entry on oeis.org

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

Offset: 0

Paul Barry, Jul 26 2005

a(n) = A162699(n+1) (Modd 7) = A204453(A162699(n+1)), n>=0, where the nonnegative members of the seven residue classes Mod 7 (not to be confused with mod 7), called [m] for m=0..6, are given in the array A113807, if there the last row, starting with 7 is taken as class [0] after adding a 0 in front. Here only the classes [1], [3] and [5] are relevant. For Modd n residue classes see a comment on A203571. [Wolfdieter Lang, Feb 09 2012]

Continued fractions expansion of (8+sqrt(905))/29 = 1.3132144107925.. - R. J. Mathar, Mar 08 2012

			Modd 7 classes for positive odd numbers reduced mod 7: a(3)=5 because A162699(4)=9 (the fourth positive odd number not divisible by 7), and 9 is a member of the Modd 7 class [5] = {5,9,19,23,...}.
A162699: 1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25, 27,...
Modd 7:  1, 3, 5, 5,  3,  1,  1,  3,  5,  5,  3,  1,... [_Wolfdieter Lang_, Feb 09 2012]

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

Cf. A001045, A105198, A110549, A110550, A113807, A162699, A203571, A204453.

&cat [[1, 3, 5, 5, 3, 1]^^30]; // Wesley Ivan Hurt, Jun 29 2016

A110551:=n->[1, 3, 5, 5, 3, 1][(n mod 6)+1]: seq(A110551(n), n=0..100); # Wesley Ivan Hurt, Jun 29 2016

PadRight[{}, 100, {1, 3, 5, 5, 3, 1}] (* Wesley Ivan Hurt, Jun 29 2016 *)

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

From R. J. Mathar, Oct 15 2014: (Start)
G.f.: ( 1+x+x^2 ) / ( (1-x)*(x^2-x+1) ).
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = 3 + 2*sin(Pi*n/3)/sqrt(3) - 2*cos(Pi*n/3).
a(n) = A001045(n+2) mod 6. (End)
From Wesley Ivan Hurt, Jun 29 2016: (Start)
a(n) = a(n-6) for n>5.
a(n) = 2*a(n-1) - 2*a(n-2) + a(n-3) for n>2. (End)

A110550 Periodic {1,3,2,4,4,2,3,1}.

Original entry on oeis.org

1, 3, 2, 4, 4, 2, 3, 1, 1, 3, 2, 4, 4, 2, 3, 1, 1, 3, 2, 4, 4, 2, 3, 1, 1, 3, 2, 4, 4, 2, 3, 1, 1, 3, 2, 4, 4, 2, 3, 1, 1, 3, 2, 4, 4, 2, 3, 1, 1, 3, 2, 4, 4, 2, 3, 1, 1, 3, 2, 4, 4, 2, 3, 1, 1, 3, 2, 4, 4, 2, 3, 1, 1, 3, 2, 4, 4, 2, 3, 1, 1, 3, 2, 4, 4, 2, 3, 1, 1, 3, 2, 4, 4, 2, 3, 1

Offset: 0

Paul Barry, Jul 26 2005

Permutation of {1,2,3,4} followed by its reversal, repeated.

Simple continued fraction expansion of (671 + sqrt 7241477)/2606. - R. J. Mathar, Mar 08 2012

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for linear recurrences with constant coefficients, signature (1,0,0,-1,1).

Cf. A105198, A110549.

PadRight[{}, 100, {1,3,2,4,4,2,3,1}] (* G. C. Greubel, Aug 31 2017 *)

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

(define (A110550 n) (list-ref '(1 3 2 4 4 2 3 1) (modulo n 8))) ;; Antti Karttunen, Aug 10 2017

G.f.: -(x^2+3*x+1)*(x^2-x+1) / ( (x-1)*(1+x^4) ).

a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) - a(n-6) + a(n-7).

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

A111951 Period 8: repeat [0,3,1,2,2,1,3,0].

Original entry on oeis.org

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

Offset: 0

Paul Barry, Aug 22 2005

Permutation of {0,1,2,3} followed by its reversal, repeated.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1).

Cf. A110549, A110550.

(define (A111951 n) (list-ref '(0 3 1 2 2 1 3 0) (modulo n 8))) ;; Antti Karttunen, Aug 10 2017

G.f.: (3x + x^2 + 2x^3 + 2x^4 + x^5 + 3x^6)/(1 - x^8);
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) = n(7n-1)/2 mod 4 = A022264(n) mod 4.
G.f.: -x*(3 - 2*x + 4*x^2 - 2*x^3 + 3*x^4) / ( (x-1)*(1+x^2)*(1+x^4) ). - R. J. Mathar, Feb 20 2015
a(n) = (3 + r/2 - s/2 + 2*cos(Pi*(1+2*n-r-s+t)/8) - 2*cos(Pi*(1-2*n+r-s+t)/8) - 2*sin(Pi*(1-2*n-r+s+t)/8))/2 where r = 2*sin(n*Pi/2), s = 2*cos(n*Pi/2) and t = cos(n*Pi). - Wesley Ivan Hurt, Oct 05 2018

Name changed, the original name moved to comments. - Antti Karttunen, Aug 10 2017

cp's OEIS Frontend

A105198 a(n) = n(n+1)/2 mod 4.

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A110551 Period 6: repeat [1, 3, 5, 5, 3, 1].

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A110550 Periodic {1,3,2,4,4,2,3,1}.

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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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A111951 Period 8: repeat [0,3,1,2,2,1,3,0].

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