A084100 -id:A084100 - cp's OEIS frontend

A084099 Expansion of (1+x)^2/(1+x^2).

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

Offset: 0

Paul Barry, May 15 2003

Inverse binomial transform of A077860. Partial sums of A084100.
Transform of sqrt(1+2x)/sqrt(1-2x) (A063886) under the Chebyshev transformation A(x)->((1-x^2)/(1+x^2))*A(x/(1+x^2)). - Paul Barry, Oct 12 2004
Euler transform of length 4 sequence [2, -3, 0, 1]. - Michael Somos, Aug 04 2009

			G.f. = 1 + 2*x - 2*x^3 + 2*x^5 - 2*x^7 + 2*x^9 - 2*x^11 + 2*x^13 - 2*x^15 + ...

Colin Barker, Table of n, a(n) for n = 0..1000
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28.
Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A063886, A077860, A084100, A101455, A180969.

[1] cat [Integers()!((1-(-1)^n)*(-1)^(n*(n-1)/2)): n in [1..100]]; // Wesley Ivan Hurt, Oct 27 2015

A084099:=n->(1-(-1)^n)*(-1)^((2*n-1+(-1)^n)/4): 1,seq(A084099(n), n=1..100); # Wesley Ivan Hurt, Oct 27 2015

CoefficientList[Series[(1+x)^2/(1+x^2),{x,0,110}],x] (* or *) Join[ {1}, PadRight[{},120,{2,0,-2,0}]] (* Harvey P. Dale, Nov 23 2011 *)

{a(n) = if( n<1, n==0, 2 * if( n%2, (-1)^(n\2)) )}; /* Michael Somos, Aug 04 2009 */

a(n) = if(n==0, 1, I*((-I)^n-I^n)) \\ Colin Barker, Oct 27 2015

Vec((1+x)^2/(1+x^2) + O(x^100)) \\ Colin Barker, Oct 27 2015

G.f.: (1+x)^2/(1+x^2).
a(n) = 2 * A101455(n) for n>0. - N. J. A. Sloane, Jun 01 2010
a(n+2) = (-1)^A180969(1,n)*((-1)^n - 1). - Adriano Caroli, Nov 18 2010
G.f.: 4*x + 2/(1+x)/G(0), where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 19 2013
From Wesley Ivan Hurt, Oct 27 2015: (Start)
a(n) = (1-sign(n)*(-1)^n)*(-1)^floor(n/2).
a(n) = 2*(n mod 2)*(-1)^floor(n/2) for n>0, a(0)=1.
a(n) = (1-(-1)^n)*(-1)^(n*(n-1)/2) for n>0, a(0)=1. (End)
From Colin Barker, Oct 27 2015: (Start)
a(n) = -a(n-2).
a(n) = i*((-i)^n-i^n) for n>0, where i = sqrt(-1).
(End)

A280560 a(n) = (-1)^n * 2 if n!=0, with a(0) = 1.

Original entry on oeis.org

1, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2, -2, 2

Offset: 0

Michael Somos, Jan 05 2017

			G.f. = 1 - 2*x + 2*x^2 - 2*x^3 + 2*x^4 - 2*x^5 + 2*x^6 - 2*x^7 + 2*x^8 - 2*x^9 + ...

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

Cf. A033999, A040000, A084100.

[n eq 0 select 1 else 2*(-1)^n: n in [0..75]]; // G. C. Greubel, Jul 29 2018; Mar 28 2024

a[ n_] := (-1)^n (2 - Boole[n == 0]);
PadRight[{1},120,{2,-2}] (* Harvey P. Dale, Jun 04 2019 *)

PARI
```
{a(n) = (-1)^n * if(n, 2, 1)};
```

[2*(-1)^n -int(n==0) for n in range(76)] # G. C. Greubel, Mar 28 2024

Euler transform of length 2 sequence [-2, 1].
Moebius transform is length 2 sequence [-2, 4].
a(n) = -2*A033999(n) if n!=0.
G.f.: (1 - x) / (1 + x) = 1 / (1 + 2*x / (1 - x)) = 1 - 2*x / (1 + x).
E.g.f.: 2*exp(-x) - 1.
a(n) = a(-n) for all n in Z.
a(n) = A084100(2*n) = A084100(2*n + 1), if n>=0.
a(n) = (-1)^n * A040000(n).
a(2*n) = A040000(n).
Convolution inverse is A040000.

A018840 Number of steps for {2,3} fairy knight to reach (n,0) on infinite chessboard.

Original entry on oeis.org

0, 5, 4, 5, 2, 5, 2, 5, 4, 5, 4, 7, 4, 5, 6, 7, 6, 7, 6, 7, 8, 9, 8, 9, 8, 9, 10, 11, 10, 11, 10, 11, 12, 13, 12, 13, 12, 13, 14, 15, 14, 15, 14, 15, 16, 17, 16, 17, 16, 17, 18, 19, 18, 19, 18, 19, 20, 21, 20, 21, 20, 21, 22, 23, 22, 23, 22, 23, 24, 25, 24, 25, 24, 25, 26, 27, 26, 27, 26, 27

Offset: 0

Marc LeBrun

This piece is also known as a (2,3)-leaper or a zebra. - Franklin T. Adams-Watters, Dec 27 2017

Apparently also the minimum number of moves of the (1,5)-leaper to reach (n,n) starting from (0,0). - R. J. Mathar, Jan 05 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1)

concat(0, Vec(x*(5 - x + x^2 - 3*x^3 + 3*x^4 - 3*x^5 - 2*x^6 + 2*x^9 - 2*x^12 + 2*x^13 - 2*x^16 + 2*x^17) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)) + O(x^100))) \\ Colin Barker, Dec 28 2017

For n >= 18, a(n) = a(n-6) + 2. - David W. Wilson
From Colin Barker, Dec 28 2017: (Start)
G.f.: x*(5 - x + x^2 - 3*x^3 + 3*x^4 - 3*x^5 - 2*x^6 + 2*x^9 - 2*x^12 + 2*x^13 - 2*x^16 + 2*x^17) / ((1 - x)^2*(1 + x)*(1 - x + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + a(n-6) - a(n-7) for n>7.
(End)
3*a(n) = A004442(n+3)-A084100(n), n>11. - R. J. Mathar, Jan 02 2018

A216178 Period 4: repeat [4, 1, 0, -3].

Original entry on oeis.org

4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3, 4, 1, 0, -3

Offset: 0

Brad Clardy, Mar 10 2013

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

Cf. A084100, A168361, A214864.

for n in [0 .. 50] do (3*(-1)^n+1)/2 + 2*(-1)^((2*n-1+(-1)^n)/4); end for;

&cat [[4, 1, 0, -3]^^30]; // Wesley Ivan Hurt, Jul 09 2016

seq(op([4, 1, 0, -3]), n=0..40); # Wesley Ivan Hurt, Jul 09 2016

PadRight[{},100,{4,1,0,-3}] (* or *) LinearRecurrence[{0,0,0,1},{4,1,0,-3},100] (* Harvey P. Dale, Nov 28 2014 *)

a(n)=[4, 1, 0, -3][n%4+1] \\ Charles R Greathouse IV, Jul 17 2016

a(n) = (3*(-1)^n+1)/2 + 2*(-1)^((2*n-1+(-1)^n)/4).
a(n) = A168361(n+1) + A084100(n+4).
G.f.: (4+x-3*x^3) / ((1-x)*(1+x)*(1+x^2)). - R. J. Mathar, Mar 10 2013
a(n+4) = a(n). - Alexander R. Povolotsky, Mar 15 2013
From Wesley Ivan Hurt, Jul 09 2016: (Start)
a(n) = 1/2+3*I^(2*n)/2+(1+I)*I^(-n)+(1-I)*I^n.
a(n) = (1+3*cos(n*Pi)+4*cos(n*Pi/2)+4*sin(n*Pi/2)+3*I*sin(n*Pi))/2. (End)

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A084099 Expansion of (1+x)^2/(1+x^2).

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A280560 a(n) = (-1)^n * 2 if n!=0, with a(0) = 1.

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A018840 Number of steps for {2,3} fairy knight to reach (n,0) on infinite chessboard.

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A216178 Period 4: repeat [4, 1, 0, -3].

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