A105198 -id:A105198 - cp's OEIS frontend

A110551 Period 6: repeat [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

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)

A110549 Period 8: repeat [1, 2, 4, 3, 3, 4, 2, 1].

Original entry on oeis.org

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

Offset: 0

Paul Barry, Jul 26 2005

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

Continued fraction expansion of (337 + sqrt(905669))/890 = 1.44793981253727... - 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,-1,1,-1,1,-1,1).

One more than A105198.

PadRight[{},120,{1,2,4,3,3,4,2,1}] (* Harvey P. Dale, May 12 2015 *)

a(n)=[1,2,4,3,3,4,2,1][n%8+1] \\ Charles R Greathouse IV, Oct 16 2015

G.f.: (1 + x + 3*x^2 + 3*x^4 + x^5 + x^6)/(1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7). [corrected by Georg Fischer, May 15 2019]
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 + 5/2.
a(n) = 1 + A105198(n).
a(n) = 1 + (A000217(n) mod 4). - Jon E. Schoenfield, Aug 11 2017

A133882 a(n) = binomial(n+2,n) mod 2^2.

Original entry on oeis.org

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, 1

Offset: 0

Hieronymus Fischer, Oct 10 2007

Periodic with length 2^3 = 8.

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. A000040, A133620-A133625, A133630, A038509, A133634-A133636.
Cf. A133872, A133880, A133890, A133900, A133910.
For the sequence regarding "binomial(n+2, n) mod 2" see A133872.
A105198 shifted once left.

Table[Mod[Binomial[n+2,n],4],{n,0,120}]  (* Harvey P. Dale, Apr 16 2011 *)

A133882(n) = (binomial(n+2,n) % 4); \\ Antti Karttunen, Aug 10 2017

a(n) = binomial(n+2,2) mod 2^2.
G.f.: (1 + 3*x + 2*x^2 + 2*x^3 + 3*x^4 + x^5)/(1-x^8).
G.f.: (1+x)*(1+2*x+2*x^3+x^4)/(1-x^8) = (1+2*x+2*x^3+x^4)/((1-x)*(1+x^2)*(1+x^4)).
a(n) = A105198(n+1). - R. J. Mathar, Jun 08 2008

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).

A105332 a(n) = n*(n+1)/2 mod 8.

Original entry on oeis.org

0, 1, 3, 6, 2, 7, 5, 4, 4, 5, 7, 2, 6, 3, 1, 0, 0, 1, 3, 6, 2, 7, 5, 4, 4, 5, 7, 2, 6, 3, 1, 0, 0, 1, 3, 6, 2, 7, 5, 4, 4, 5, 7, 2, 6, 3, 1, 0, 0, 1, 3, 6, 2, 7, 5, 4, 4, 5, 7, 2, 6, 3, 1, 0, 0, 1, 3, 6, 2, 7, 5, 4, 4, 5, 7, 2, 6, 3, 1, 0, 0, 1, 3, 6, 2, 7, 5, 4, 4, 5, 7, 2, 6, 3, 1, 0, 0, 1, 3, 6, 2, 7, 5, 4, 4

Offset: 0

Oscar Takeshita, May 01 2005

Periodic with period length 16. - Ray Chandler, Apr 18 2025

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

Cf. A000217.

See A105198 for further information.

Mod[Accumulate[Range[0,110]],8] (* Harvey P. Dale, Nov 03 2013 *)

def A105332(n): return (n*(n+1)>>1)&7 # Chai Wah Wu, Apr 17 2025

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

A105340 a(n) = n*(n+1)/2 mod 2048.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1830, 1891, 1953, 2016, 32

Offset: 0

Oscar Takeshita, May 01 2005

Period 4096. - Charles R Greathouse IV, Oct 16 2012

Index entries for linear recurrences with constant coefficients, order 4095.

See A105198 for further information.

Cf. A000217.

Mod[Accumulate[Range[0,70]],2048] (* Harvey P. Dale, Sep 18 2019 *)

a(n)=n*(n+1)/2%2048 \\ Charles R Greathouse IV, Oct 16 2012

def A105340(n): return (n*(n+1)>>1)&2047 # Chai Wah Wu, Apr 17 2025

Removed formulas that were wrong at indices >= 65 because the "mod 2048" part of the definition had been ignored R. J. Mathar, Jan 26 2010

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

A053794 a(n) = (n^2 + n) modulo 8.

Original entry on oeis.org

0, 2, 6, 4, 4, 6, 2, 0, 0, 2, 6, 4, 4, 6, 2, 0, 0, 2, 6, 4, 4, 6, 2, 0, 0, 2, 6, 4, 4, 6, 2, 0, 0, 2, 6, 4, 4, 6, 2, 0, 0, 2, 6, 4, 4, 6, 2, 0, 0, 2, 6, 4, 4, 6, 2, 0, 0, 2, 6, 4, 4, 6, 2, 0, 0, 2, 6, 4, 4, 6, 2, 0, 0, 2, 6, 4, 4, 6, 2, 0, 0, 2, 6, 4, 4, 6, 2, 0, 0, 2, 6, 4, 4, 6, 2, 0, 0, 2, 6, 4, 4, 6, 2, 0

Offset: 0

Stuart M. Ellerstein (ellerstein(AT)aol.com), Mar 27 2000

Periodic with period 8.

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. A053793, A105198.

PadRight[{}, 105, {0, 2, 6, 4, 4, 6, 2, 0}] (* Michael De Vlieger, Jun 22 2022 *)

A053794(n)=[0, 2, 6, 4, 4, 6, 2, 0][n%8+1] \\ M. F. Hasler, Aug 27 2012

(define (A053794 n) (modulo (+ (* n n) n) 8)) ;; Antti Karttunen, Aug 10 2017

From Wesley Ivan Hurt, Jun 22 2022: (Start)
a(n) = 2*A105198(n).
a(n) = a(n-1)-a(n-2)+a(n-3)-a(n-4)+a(n-5)-a(n-6)+a(n-7). (End)

More terms from James Sellers, Apr 08 2000

Extended to n=103 by Antti Karttunen, Aug 10 2017

A105333 a(n) = n*(n+1)/2 mod 16.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 5, 12, 4, 13, 7, 2, 14, 11, 9, 8, 8, 9, 11, 14, 2, 7, 13, 4, 12, 5, 15, 10, 6, 3, 1, 0, 0, 1, 3, 6, 10, 15, 5, 12, 4, 13, 7, 2, 14, 11, 9, 8, 8, 9, 11, 14, 2, 7, 13, 4, 12, 5, 15, 10, 6, 3, 1, 0, 0, 1, 3, 6, 10, 15, 5, 12, 4, 13, 7, 2, 14, 11, 9, 8, 8, 9, 11, 14, 2, 7, 13, 4, 12

Offset: 0

Oscar Takeshita, May 01 2005

nonn

Triangular numbers mod 16. - Harvey P. Dale, Oct 12 2012

Periodic with period length 32. - Ray Chandler, Apr 18 2025

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

Cf. A000217.

See A105198 for further information.

Mod[#,16]&/@Accumulate[Range[0,90]] (* Harvey P. Dale, Oct 12 2012 *)

def A105333(n): return (n*(n+1)>>1)&15 # Chai Wah Wu, Apr 17 2025

From Chai Wah Wu, Apr 17 2025: (Start)
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) + a(n-9) - a(n-10) + a(n-11) - a(n-12) + a(n-13) - a(n-14) + a(n-15) - a(n-16) + a(n-17) - a(n-18) + a(n-19) - a(n-20) + a(n-21) - a(n-22) + a(n-23) - a(n-24) + a(n-25) - a(n-26) + a(n-27) - a(n-28) + a(n-29) - a(n-30) + a(n-31) for n > 30.
G.f.: x*(-x^28 - 2*x^27 - 4*x^26 - 6*x^25 - 9*x^24 + 4*x^23 - 16*x^22 + 12*x^21 - 25*x^20 + 18*x^19 - 20*x^18 + 6*x^17 - 17*x^16 + 8*x^15 - 16*x^14 + 8*x^13 - 17*x^12 + 6*x^11 - 20*x^10 + 18*x^9 - 25*x^8 + 12*x^7 - 16*x^6 + 4*x^5 - 9*x^4 - 6*x^3 - 4*x^2 - 2*x - 1)/((x - 1)*(x^2 + 1)*(x^4 + 1)*(x^8 + 1)*(x^16 + 1)). (End)

cp's OEIS Frontend

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

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A110549 Period 8: repeat [1, 2, 4, 3, 3, 4, 2, 1].

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A133882 a(n) = binomial(n+2,n) mod 2^2.

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

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A105332 a(n) = n*(n+1)/2 mod 8.

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

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A105340 a(n) = n*(n+1)/2 mod 2048.

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