A068073 -id:A068073 - cp's OEIS frontend

A028356 Simple periodic sequence underlying clock sequence A028354.

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

Offset: 0

N. J. A. Sloane

From Klaus Brockhaus, May 15 2010: (Start)
Continued fraction expansion of (28+sqrt(2730))/56.
Decimal expansion of 1112/9009.
Partial sums of 1 followed by A130151.
First differences of A028357. (End)

Zdeněk Horský, "Pražský orloj" ("The Astronomical Clock of Prague", in Czech), Panorama, Prague, 1988, pp. 76-78.

Michal Křížek, Alena Šolcová and Lawrence Somer, Construction of Šindel sequences, Comment. Math. Univ. Carolin., 48 (2007), 373-388.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A000034, A028354, A068073, A118382, A118383.

Cf. A177924 (decimal expansion of (28+sqrt(2730))/56), A130151 (repeat 1, 1, 1, -1, -1, -1), A028357 (partial sums of A028356). - Klaus Brockhaus, May 15 2010

&cat [[1, 2, 3, 4, 3, 2]^^20]; // Klaus Brockhaus, May 15 2010

A028356:=n->[1, 2, 3, 4, 3, 2][(n mod 6)+1]: seq(A028356(n), n=0..100); # Wesley Ivan Hurt, Jun 23 2016

CoefficientList[ Series[(1 + 2x + 3x^2 + 4x^3 + 3x^4 + 2x^5)/(1 - x^6), {x, 0, 85}], x]
LinearRecurrence[{1,0,-1,1},{1,2,3,4},120] (* or *) PadRight[{},120,{1,2,3,4,3,2}] (* Harvey P. Dale, Apr 15 2016 *)

def A028356(n): return (1,2,3,4,3,2)[n%6] # Chai Wah Wu, Apr 18 2024

def A():
    a, b, c, d = 1, 2, 3, 4
    while True:
        yield a
        a, b, c, d = b, c, d, a + (d - b)
A028356 = A(); [next(A028356) for n in range(106)] # Peter Luschny, Jul 26 2014

Sum of any six successive terms is 15.
G.f.: (1 + 2*x + 3*x^2 + 4*x^3 + 3*x^4 + 2*x^5)/(1 - x^6).
From Wesley Ivan Hurt, Jun 23 2016: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
a(n) = (15 - cos(n*Pi) - 8*cos(n*Pi/3))/6. (End)
E.g.f.: (15*exp(x) - exp(-x) - 8*cos(sqrt(3)*x/2)*(sinh(x/2) + cosh(x/2)))/6. - Ilya Gutkovskiy, Jun 23 2016
a(n) = abs(((n+3) mod 6)-3) + 1. - Daniel Jiménez, Jan 14 2023

Additional comments from Robert G. Wilson v, Mar 01 2002

A158289 Period 18 zigzag sequence: repeat [0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1].

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5

Offset: 0

Jaroslav Krizek, Mar 15 2009

A toothed or zigzag sequence.
Sequence contains only numbers 0..9; abs(a(n+1)-a(n)) = 1.
Decimal expansion of 12345679/1000000001. - Elmo R. Oliveira, Feb 20 2024

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,-1,1).

Cf. A068073 (repeat 1,2,3,2), A028356 (repeat 1,2,3,4,3,2), A130784 (repeat 1,3,2).

Period k zigzag sequences: A000035 (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), this sequence (k=18).

[ s lt 9 select r else 9-r where r is n mod 9 where s is n mod 18: n in [0..104] ]; // Klaus Brockhaus, Sep 07 2009

S:=[]; a:=0; for n in [0..104] do Append(~S, a); if n mod 18 eq 0 then d:=1; else if n mod 9 eq 0 then d:=-1; end if; end if; a+:=d; end for; S; // Klaus Brockhaus, Sep 07 2009

&cat[[0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1]: n in [0..5]]; // Vincenzo Librandi, Jul 26 2015

a[n_] := If[m = Mod[n, 18]; m <= 9, m, 18-m]; Table[a[n], {n, 0, 85}] (* Jean-François Alcover, Jul 19 2013 *)
PadRight[{}, 100, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1}] (* Vincenzo Librandi, Jul 26 2015 *)

a(n)=abs(n-round(n/18)*18) \\ M. F. Hasler, Jul 27 2015

a(18*k+j) = a(18*(k+1)-j) = j for k >= 0, j = 0..9.
G.f.: x*(1+x+x^2)*(1+x^3+x^6)/((1-x)*(1+x)*(1-x+x^2)*(1-x^3+x^6)). - Klaus Brockhaus, Sep 07 2009
a(n) = Sum_{i=0..n-1} (-1)^floor(i/9). - Wesley Ivan Hurt, Jul 25 2015
a(n) = abs(n - 18*round(n/18)). - Wesley Ivan Hurt, Dec 10 2016
a(n) = a(n-18) for n >= 18. - Wesley Ivan Hurt, Sep 07 2022

Edited and extended by Klaus Brockhaus, Sep 07 2009

A186111 a(n) = -n if n odd, a(2n) = 3n if n odd, a(4n) = 2n.

Original entry on oeis.org

1, -3, 3, -2, 5, -9, 7, -4, 9, -15, 11, -6, 13, -21, 15, -8, 17, -27, 19, -10, 21, -33, 23, -12, 25, -39, 27, -14, 29, -45, 31, -16, 33, -51, 35, -18, 37, -57, 39, -20, 41, -63, 43, -22, 45, -69, 47, -24, 49, -75, 51, -26, 53, -81, 55, -28, 57, -87, 59, -30, 61, -93, 63

Offset: 1

Michael Somos, Feb 13 2011

			G.f. = x - 3*x^2 + 3*x^3 - 2*x^4 + 5*x^5 - 9*x^6 + 7*x^7 - 4*x^8 + 9*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Rational Function Multiplicative Coefficients.
Index entries for linear recurrences with constant coefficients, signature (-2,-3,-4,-3,-2,-1).

Cf. A068073, A186690, A186813.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-x)^3*(1-x^3)/(1-x^4)^2)); // G. C. Greubel, Aug 14 2018

Rest[CoefficientList[Series[x (1-x)^3(1-x^3)/(1-x^4)^2,{x,0,70}],x]] (* or *) LinearRecurrence[{-2,-3,-4,-3,-2,-1},{1,-3,3,-2,5,-9},70] (* Harvey P. Dale, Aug 08 2012 *)
a[ n_] := n If[ OddQ[n], 1, -(Mod[n/2, 2] + 1/2)]; (* Michael Somos, Apr 25 2015 *)
a[ n_] := n {1, -3/2, 1, -1/2}[[Mod[n, 4, 1]]]; (* Michael Somos, Apr 25 2015 *)

{a(n) = -(-1)^n * n * [1, 2, 3, 2] [n%4 + 1] / 2};

{a(n) = sign(n) * polcoeff( x * (1 - x)^3 * (1 - x^3) / (1 - x^4)^2 + x * O(x^abs(n)), abs(n))};

{a(n) = n * if( n%2, 1, -(n/2%2 + 1/2))}; /* Michael Somos, Apr 25 2015 */

a(n) is multiplicative with a(2) = -3, a(2^e) = -(2^(e-1)) if e>1, a(p^e) = p^e if p>2.
Euler transform of length 4 sequence [-3, 0, -1, 2].
G.f.: x * (1 - x)^3 * (1 - x^3) / (1 - x^4)^2.
G.f.: x * (1 + x + x^2) * (1 - x)^2 / ((1 + x)^2 * (1 + x^2)^2).
Dirichlet g.f. zeta(s-1)*( 1-5*2^(-s)+4^(1-s)). - R. J. Mathar, Mar 31 2011
a(n) = (-1)^(n+1)*n + (-1)^floor(n/2)*A027656(n-2). - R. J. Mathar, Mar 31 2011
a(n) = -2*a(n-1) - 3*a(n-2) - 4*a(n-3) - 3*a(n-4) - 2*a(n-5) - a(n-6) with a(1)=1, a(2)=-3, a(3)=3, a(4)=-2, a(5)=5, a(6)=-9. - Harvey P. Dale, Aug 08 2012
G.f.: 1/(1+x) - 1/(1+x)^2 - 1/(1+x^2) + 1/(1+x^2)^2. - Michael Somos, Apr 24 2015
a(n) = -a(-n) for all n in Z. - Michael Somos, Apr 24 2015
G.f.: f(x) - f(x^2) where f(x) := x / (1 + x)^2. - Michael Somos, May 07 2015
Moebius transform of A186690. - Michael Somos, Apr 25 2015
a(n) = -(-1)^n * A186813(n). - Michael Somos, May 07 2015
a(n) = n*cos(n*Pi/2)/2-n*(-1)^n. - Wesley Ivan Hurt, May 05 2021

A186813 a(n) = n if n odd, a(2n) = 3n if n odd, a(4n) = 2n.

Original entry on oeis.org

0, 1, 3, 3, 2, 5, 9, 7, 4, 9, 15, 11, 6, 13, 21, 15, 8, 17, 27, 19, 10, 21, 33, 23, 12, 25, 39, 27, 14, 29, 45, 31, 16, 33, 51, 35, 18, 37, 57, 39, 20, 41, 63, 43, 22, 45, 69, 47, 24, 49, 75, 51, 26, 53, 81, 55, 28, 57, 87, 59, 30, 61, 93, 63, 32, 65, 99, 67, 34, 69, 105, 71, 36

Offset: 0

Michael Somos, Feb 27 2011

			G.f. = x + 3*x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 9*x^6 + 7*x^7 + 4*x^8 + 9*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).

Cf. A068073, A134172, A186111.

Cf. A187601. - Bruno Berselli, Mar 12 2011

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+x)*(1+x^3)/((1-x)*(1+x^2))^2)); // G. C. Greubel, Aug 14 2018

CoefficientList[Series[x(1+x)(1+x^3)/((1-x)(1+x^2))^2,{x,0,80}],x] (* Harvey P. Dale, Mar 06 2011 *)
a[ n_] := n/2 {2, 3, 2, 1}[[ Mod[ n, 4, 1]]]; (* Michael Somos, May 04 2015 *)

PARI
```
{a(n) = n/2 * [1, 2, 3, 2][n%4 + 1]};
```

{a(n) = sign(n) * polcoeff( x * (1 + x) * (1 + x^3) / ((1 - x) * (1 + x^2))^2 + x * O(x^abs(n)), abs(n))};

a(n) is multiplicative with a(2) = 3, a(2^e) = 2^(e-1) if e>1, a(p^e) = p^e if p>2.
Euler transform of length 6 sequence [3, -3, 1, 2, 0, -1].
G.f.: x * (1 + x) * (1 + x^3) / ((1 - x) * (1 + x^2))^2.
G.f.: x * (1 - x^2)^3 * (1 - x^6) / ((1 - x)^3 * (1 - x^3) * (1 - x^4)^2). - Michael Somos, May 04 2015
G.f.: f(x) - f(-x^2) where f(x) := x/(1-x)^2. - Michael Somos, May 04 2015
a(n) = -a(-n) for all n in Z. a(n) = n/2 * A068073(n).
a(n) = n*(4-i^n-(-i)^n)/4 with i=sqrt(-1). - Bruno Berselli, Mar 10 2011
a(n) = A134172(n) + A134172(n+1). - Michael Somos, May 04 2015
a(n) = -(-1)^n * A186111(n). - Michael Somos, May 07 2015
a(n) = n - n*cos(n*Pi/2)/2. - Wesley Ivan Hurt, May 05 2021
Dirichlet g.f.: zeta(s-1) * (1 + 1/2^s - 1/4^(s-1)). - Amiram Eldar, Oct 26 2023

A292576 Permutation of the natural numbers partitioned into quadruples [4k-1, 4k-3, 4k-2, 4k], k > 0.

Original entry on oeis.org

3, 1, 2, 4, 7, 5, 6, 8, 11, 9, 10, 12, 15, 13, 14, 16, 19, 17, 18, 20, 23, 21, 22, 24, 27, 25, 26, 28, 31, 29, 30, 32, 35, 33, 34, 36, 39, 37, 38, 40, 43, 41, 42, 44, 47, 45, 46, 48, 51, 49, 50, 52, 55, 53, 54, 56, 59, 57, 58, 60, 63, 61, 62

Offset: 1

Guenther Schrack, Sep 19 2017

nonn

Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the second and third elements, then swap the first and the second element; repeat for all quadruples.

Guenther Schrack, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Index entries for sequences that are permutations of the natural numbers

Inverse: A056699(n+1) - 1 for n > 0.
Sequence of fixed points: A008586(n) for n > 0.
Subsequences:
   elements with odd index: A042964(A103889(n)) for n > 0.
   elements with even index: A042948(n) for n > 0.
   odd elements: A166519(n) for n>0.
   indices of odd elements: A042963(n) for n > 0.
   even elements: A005843(n) for n>0.
   indices of even elements: A014601(n) for n > 0.
Sum of pairs of elements:
   a(n+2) + a(n) = A163980(n+1) = A168277(n+2) for n > 0.
Difference between pairs of elements:
   a(n+2) - a(n) = (-1)^A011765(n+3)*A091084(n+1) for n > 0.
Compound relations:
   a(n) = A284307(n+1) - 1 for n > 0.
   a(n+2) - 2*a(n+1) + a(n) = (-1)^A011765(n)*A132400(n+1) for n > 0.
Compositions:
   a(n) = A116966(A080412(n)) for n > 0.
   a(A284307(n)) = A256008(n) for n > 0.
   a(A042963(n)) = A166519(n-1) for n > 0.
   A256008(a(n)) = A056699(n) for n > 0.

a = [3 1 2 4]; % Generate b-file
max = 10000;
for n := 5:max
   a(n) = a(n-4) + 4;
end;

for(n=1, 10000, print1(n + ((-1)^(n*(n-1)/2)*(2 - (-1)^n) - (-1)^n)/2, ", "))

a(1)=3, a(2)=1, a(3)=2, a(4)=4, a(n) = a(n-4) + 4 for n > 4.
O.g.f.: (2*x^3 + x^2 - 2*x + 3)/(x^5 - x^4 - x + 1).
a(n) = n + ((-1)^(n*(n-1)/2)*(2-(-1)^n) - (-1)^n)/2.
a(n) = n + (cos(n*Pi/2) - cos(n*Pi) + 3*sin(n*Pi/2))/2.
a(n) = n + n mod 2 + (ceiling(n/2)) mod 2 - 2*(floor(n/2) mod 2).
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
First Differences, periodic: (-2, 1, 2, 3), repeat; also (-1)^A130569(n)*A068073(n+2) for n > 0.

A047401 Numbers that are congruent to {0, 1, 3, 6} mod 8.

Original entry on oeis.org

0, 1, 3, 6, 8, 9, 11, 14, 16, 17, 19, 22, 24, 25, 27, 30, 32, 33, 35, 38, 40, 41, 43, 46, 48, 49, 51, 54, 56, 57, 59, 62, 64, 65, 67, 70, 72, 73, 75, 78, 80, 81, 83, 86, 88, 89, 91, 94, 96, 97, 99, 102, 104, 105, 107, 110, 112, 113, 115, 118, 120, 121, 123

Offset: 1

N. J. A. Sloane

Partial sums of A068073. - Jeremy Gardiner, Jul 20 2013.

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

Cf. A047452, A047470, A068073.

[n : n in [0..150] | n mod 8 in [0, 1, 3, 6]]; // Wesley Ivan Hurt, Jun 01 2016

A047401:=n->2*(n-1)+(I^(n*(n-1))-1)/2: seq(A047401(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016

Select[Range[0,107], MemberQ[{0, 1, 3, 6}, Mod[#, 8]]&] (* Bruno Berselli, Dec 05 2011 *)

makelist(2*(n-1)+(%i^(n*(n-1))-1)/2,n,1,55); /* Bruno Berselli, Dec 05 2011 */

my(x='x+O('x^100)); concat(0, Vec(x^2*(1+x+2*x^2)/((x^2+1)*(x-1)^2))) \\ Altug Alkan, Jun 02 2016

G.f.: x^2*(1+x+2*x^2) / ( (x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 05 2011
a(n) = 2*(n-1)+(i^(n*(n-1))-1)/2, where i=sqrt(-1). - Bruno Berselli, Dec 05 2011
From Wesley Ivan Hurt, Jun 01 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(2k) = A047452(k), a(2k-1) = A047470(k). (End)
Sum_{n>=2} (-1)^n/a(n) = Pi/16 + (3-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4. - Amiram Eldar, Dec 20 2021

A164356 Expansion of (1 - x^2)^4 / ((1 - x)^4 * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2

Offset: 0

Michael Somos, Aug 13 2009

			G.f. = 1 + 4*x + 6*x^2 + 4*x^3 + 2*x^4 + 4*x^5 + 6*x^6 + 4*x^7 + 2*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (1,-1,1).

Cf. A068073.

a[ n_] := -Boole[n == 0] + 4 - If[ EvenQ[n], (-1)^(n/2) 2, 0]; (* Michael Somos, Apr 17 2015 *)
a[ n_] := SeriesCoefficient[ -1 + 4/(1 - x) - 2/(1 + x^2), {x, 0, Abs@n}]; (* Michael Somos, Jan 07 2019 *)
LinearRecurrence[{1,-1,1},{1,4,6,4},120] (* or *) PadRight[{1},120,{2,4,6,4}] (* Harvey P. Dale, Aug 30 2024 *)

{a(n) = -(n==0) + 4 - if( n%2 == 0, (-1)^(n/2) * 2, 0)};

Euler transform of length 4 sequence [4, -4, 0, 1].
Moebius transform is length 4 sequence [4, 2, 0, -4].
a(n) = 4 * b(n) unless n=0 and b(n) is multiplicative with b(2) = 3/2, b(2^e) = 1/2 if e>1, b(p^e) = 1 if p>2.
a(n) = a(-n) for all n in Z. a(n+4) = a(n) unless n=0 or n=-4. a(2*n + 1) = 4. a(4*n) = 2 unless n=0. a(4*n + 2) = 6.
G.f.: -1 + 4 / (1 - x) - 2 / (1 + x^2).
a(n) = 2 * A068073(n) unless n=0. - Michael Somos, Apr 17 2015

A177033 Decimal expansion of (2+sqrt(14))/4.

Original entry on oeis.org

1, 4, 3, 5, 4, 1, 4, 3, 4, 6, 6, 9, 3, 4, 8, 5, 3, 4, 6, 3, 9, 5, 9, 3, 7, 1, 8, 3, 0, 7, 9, 1, 3, 7, 3, 2, 5, 4, 3, 9, 0, 0, 4, 9, 5, 1, 9, 4, 4, 6, 8, 1, 7, 3, 6, 5, 7, 5, 9, 3, 6, 3, 6, 6, 8, 3, 0, 0, 0, 8, 7, 8, 9, 0, 7, 6, 7, 3, 4, 7, 5, 6, 9, 9, 4, 2, 0, 2, 4, 7, 3, 7, 9, 8, 5, 9, 4, 8, 9, 2, 8, 7, 5, 2, 4

Offset: 1

Klaus Brockhaus, May 01 2010

Continued fraction expansion of (2+sqrt(14))/4 is A068073.

			(2+sqrt(14))/4 = 1.43541434669348534639...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010471 (decimal expansion of sqrt(14)), A068073 (repeat 1, 2, 3, 2).

RealDigits[(2+Sqrt[14])/4,10,120][[1]] (* Harvey P. Dale, Jul 18 2011 *)

A298364 Permutation of the natural numbers partitioned into quadruples [4k-2, 4k-1, 4k-3, 4k] for k > 0.

Original entry on oeis.org

2, 3, 1, 4, 6, 7, 5, 8, 10, 11, 9, 12, 14, 15, 13, 16, 18, 19, 17, 20, 22, 23, 21, 24, 26, 27, 25, 28, 30, 31, 29, 32, 34, 35, 33, 36, 38, 39, 37, 40, 42, 43, 41, 44, 46, 47, 45, 48, 50, 51, 49, 52, 54, 55, 53, 56, 58, 59, 57, 60, 62, 63, 61, 64, 66, 67, 65

Offset: 1

Guenther Schrack, Jan 18 2018

Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the first and second elements, then swap the second and third elements; repeat for all quadruples.

Guenther Schrack, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Inverse: A292576.
Sequence of fixed points: A008586(n) for n > 0.
First differences: (-1)^floor(n^2/4)*A068073(n-1) for n > 0.
Subsequences:
  elements with odd index: A042963(A103889(n)) for n > 0.
  elements with even index A014601(n) for n > 0.
  odd elements: A166519(n-1) for n > 0.
  indices of odd elements: A042964(n) for n > 0.
  even elements: A005843(n) for n > 0.
  indices of even elements: A042948(n) for n > 0.
Other similar permutations: A116966, A284307, A292576.

a = [2 3 1 4];
max = 10000;    % Generation of b-file.
for n := 5:max
   a(n) = a(n-4) + 4;
end;

Nest[Append[#, #[[-4]] + 4] &, {2, 3, 1, 4}, 63] (* or *)
Array[# + ((-1)^# + ((-1)^(# (# - 1)/2)) (1 - 2 (-1)^#))/2 &, 67] (* Michael De Vlieger, Jan 23 2018 *)
LinearRecurrence[{1,0,0,1,-1},{2,3,1,4,6},70] (* Harvey P. Dale, Dec 12 2018 *)

for(n=1, 100, print1(n + ((-1)^n + ((-1)^(n*(n-1)/2))*(1 - 2*(-1)^n))/2, ", "))

O.g.f.: (3*x^3 - 2*x^2 + x + 2)/(x^5 - x^4 - x - 1).
a(1) = 2, a(2) = 3, a(3) = 1, a(4) = 4, a(n) = a(n-4) + 4 for n > 4.
a(n) = n + ((-1)^n + ((-1)^(n*(n-1)/2))*(1 - 2*(-1)^n))/2.
a(n) = n + (cos(n*Pi) - cos(n*Pi/2) + 3*sin(n*Pi/2))/2.
a(n) = 2*floor((n+1)/2) - 4*floor((n+1)/4) + floor(n/2) + 2*floor(n/4).
a(n) = n + (-1)^floor((n-1)^2/4)*A140081(n) for n > 0.
a(n) = A056699(n+1) - 1, n > 0.
a(n+2) = A168269(n+1) - a(n), n > 0.
a(n+2) = a(n) + (-1)^floor((n+1)^2/4)*A132400(n+2) for n > 0.
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
First differences: periodic, (1, -2, 3, 2) repeat.
Compositions:
  a(n) = A080412(A116966(n-1)) for n > 0.
  a(n) = A284307(A256008(n)) for n > 0.
  a(A067060(n)) = A133256(n) for n > 0.
  A116966(a(n+1)-1) = A092486(n) for n >= 0.
  A056699(a(n)) = A256008(n) for n > 0.

A164360 Period 3: repeat [5, 4, 3].

Original entry on oeis.org

5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3

Offset: 0

Stephen Crowley, Aug 14 2009

From Klaus Brockhaus, May 29 2010: (Start)
Continued fraction expansion of (32+sqrt(1297))/13.
Decimal expansion of 181/333. (End)

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

Cf. A007877 (repeat 0,1,2,1), A068073 (repeat 1,2,3,2), A028356 (repeat 1,2,3,4,3,2), A130784 (repeat 1,3,2), A158289 (repeat 0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1).

Cf. A178566 (decimal expansion of (32+sqrt(1297))/13). [Klaus Brockhaus, May 29 2010]

[ n le 3 select 6-n else Self(n-3):n in [1..105] ]; // Klaus Brockhaus, Sep 17 2009

&cat [[5, 4, 3]^^30]; // Wesley Ivan Hurt, Jul 01 2016

seq(op([5, 4, 3]), n=0..50); # Wesley Ivan Hurt, Jul 01 2016

PadRight[{}, 100, {5, 4, 3}] (* Wesley Ivan Hurt, Jul 01 2016 *)

a(n) = 4+(-1)^n*((1/2+I*sqrt(3)/6)*((1+I*sqrt(3))/2)^n+(1/2-I*sqrt(3)/6)*((1-I*sqrt(3))/2)^n). [Corrected by Klaus Brockhaus, Sep 17 2009]
a(n) = 4+(1/3)*sqrt(3)*sin(2*n*Pi/3)+cos(2*n*Pi/3). [Corrected by Klaus Brockhaus, Sep 17 2009]
a(n) = a(n-3) for n > 2, with a(0) = 5, a(1) = 4, a(2) = 3.
G.f.: (5+4*x+3*x^2)/((1-x)*(1+x+x^2)). [Klaus Brockhaus, Sep 17 2009]
E.g.f.: 4*exp(x)+(1/3)*sqrt(3)*exp(-(1/2)*x)*sin((1/2)*x*sqrt(3))+exp(-(1/2)*x)*cos((1/2)*x*sqrt(3)).
a(n) = 4 + A057078(n). - Wesley Ivan Hurt, Jul 01 2016

Edited by Klaus Brockhaus, Sep 17 2009

Offset changed to 0 and formulas adjusted by Klaus Brockhaus, May 18 2010

cp's OEIS Frontend

A028356 Simple periodic sequence underlying clock sequence A028354.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Python

Sage

Formula

Extensions

A158289 Period 18 zigzag sequence: repeat [0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Magma

Magma

Mathematica

PARI

Formula

Extensions

A186111 a(n) = -n if n odd, a(2n) = 3n if n odd, a(4n) = 2n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Formula

A186813 a(n) = n if n odd, a(2n) = 3n if n odd, a(4n) = 2n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A292576 Permutation of the natural numbers partitioned into quadruples [4k-1, 4k-3, 4k-2, 4k], k > 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

MATLAB

PARI

Formula

A047401 Numbers that are congruent to {0, 1, 3, 6} mod 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple