A075553 -id:A075553 - cp's OEIS frontend

A166486 Periodic sequence [0,1,1,1] of length 4; Characteristic function of numbers that are not multiples of 4.

0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0

Offset: 0

Jaume Oliver Lafont, Oct 15 2009

			G.f. = x + x^2 + x^3 + x^5 + x^6 + x^7 + x^9 + x^10 + x^11 + x^13 + x^14 + ...

Antti Karttunen, Table of n, a(n) for n = 0..65537
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for characteristic functions
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Characteristic function of A042968, whose complement A008586 gives the positions of zeros (after its initial term).
Absolute values of A046978, A075553, A131729, A358839, and for n >= 1, also of A112299 and of A257196.
Sequence A152822 shifted by two terms.
Parity of A327860, A331733, A342002, A351083 and A345000.
Row 3 of A225145, Column 2 of A229940 (after the initial term).
First differences of A057353. Sum of A359370 and A359372.
Cf. A000035, A011655, A011558, A097325, A109720, A168181, A168182, A168184, A145568, A168185 (characteristic functions for numbers that are not multiples of k = 2, 3 and 5..12).
Cf. A016628, A164985, A165132.
Cf. A010873, A033436, A069733 (inverse Möbius transform), A121262 (one's complement), A190621 [= n*a(n)], A355689 (Dirichlet inverse).

[Ceiling(n/4)-Floor(n/4) : n in [0..50]]; // Wesley Ivan Hurt, Jun 20 2014

seq(1/2*((n^3+n) mod 4), n=0..50); # Gary Detlefs, Mar 20 2010

PadRight[{},120,{0,1,1,1}] (* Harvey P. Dale, Jul 04 2013 *)
Table[Ceiling[n/4] - Floor[n/4], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 20 2014 *)
a[ n_] := Sign[ Mod[n, 4]]; (* Michael Somos, May 05 2015 *)

PARI
```
{a(n) = !!(n%4)};
```

def A166486(n): return (0,1,1,1)[n&3] # Chai Wah Wu, Jan 03 2023

G.f.: (x + x^2 + x^3) / (1 - x^4) = x * (1 + x + x^2) / ((1 - x) * (1 + x) * (1 + x^2)) = x * (1 - x^3) / ((1 - x) * (1 - x^4)).
a(n) = (3 - i^n - (-i)^n - (-1)^n) / 4, where i=sqrt(-1).
Sum_{k>0} a(k)/(k*3^k) = log(5)/4.
From Reinhard Zumkeller, Nov 30 2009: (Start)
Multiplicative with a(p^e) = (if p=2 then 0^(e-1) else 1), p prime and e>0.
a(n) = 1-A121262(n).
a(A042968(n))=1; a(A008586(n))=0.
A033436(n) = Sum{k=0..n} a(k)*(n-k). (End)
a(n) = 1/2*((n^3+n) mod 4). - Gary Detlefs, Mar 20 2010
a(n) = (Fibonacci(n)*Fibonacci(3n) mod 3)/2. - Gary Detlefs Dec 21 2010
Euler transform of length 4 sequence [ 1, 0, -1, 1]. - Michael Somos, Feb 12 2011
Dirichlet g.f. (1-1/4^s)*zeta(s). - R. J. Mathar, Feb 19 2011
a(n) = Fibonacci(n)^2 mod 3. - Gary Detlefs, May 16 2011
a(n) = -1/4*cos(Pi*n)-1/2*cos(1/2*Pi*n)+3/4. - Leonid Bedratyuk, May 13 2012
For the general case: the characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013
a(n) = ceiling(n/4) - floor(n/4). - Wesley Ivan Hurt, Jun 20 2014
a(n) = a(-n) for all n in Z. - Michael Somos, May 05 2015
For n >= 1, a(n) = A053866(A225546(n)) = A000035(A331733(n)). - Antti Karttunen, Jul 07 2020
a(n) = signum(n mod 4). - Alois P. Heinz, May 12 2021
From Antti Karttunen, Dec 28 2022: (Start)
a(n) = [A010873(n) > 0], where [ ] is the Iverson bracket.
a(n) = abs(A046978(n)) = abs(A075553(n)) = abs(A131729(n)) = abs(A358839(n)).
For all n >= 1, a(n) = abs(A112299(n)) = abs(A257196(n))
a(n) = A152822(2+n).
a(n) = A359370(n) + A359372(n). (End)
E.g.f.: (cosh(x) - cos(x))/2 + sinh(x). - Stefano Spezia, Aug 04 2025

Secondary definition (from Reinhard Zumkeller's Nov 30 2009 comment) added to the name by Antti Karttunen, Dec 20 2022

A009116 Expansion of e.g.f. cos(x) / exp(x).

Original entry on oeis.org

1, -1, 0, 2, -4, 4, 0, -8, 16, -16, 0, 32, -64, 64, 0, -128, 256, -256, 0, 512, -1024, 1024, 0, -2048, 4096, -4096, 0, 8192, -16384, 16384, 0, -32768, 65536, -65536, 0, 131072, -262144, 262144, 0, -524288, 1048576, -1048576, 0, 2097152, -4194304

Offset: 0

R. H. Hardin

Apart from signs, generated by 1,1 position of H_2^n = [1,1;-1,1]^n; and a(n) = 2^(n/2)*cos(Pi*n/2). - Paul Barry, Feb 18 2004
Equals binomial transform of "Period 4, repeat [1, 0, -1, 0]". - Gary W. Adamson, Mar 25 2009
Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - R. J. Mathar, Aug 10 2012

			G.f. = 1 - x + 2*x^3 - 4*x^4 + 4*x^5 - 8*x^7 + 16*x^8 - 16*x^9 + 32*x^11 - 64*x^12 + ...

N. J. A. Sloane, Table of n, a(n) for n = 0..2000
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
N. J. A. Sloane, Table of n, (I-1)^n for n=0..100
Index entries for linear recurrences with constant coefficients, signature (-2,-2).

Cf. A009545, A016116, A075553, A090132, A098158, A099087, A146559.
(With different signs) row sums of triangle A104597.
Also related to A066321 and A271472.

R:=PowerSeriesRing(Rationals(), 50); Coefficients(R!(Laplace( Exp(-x)*Cos(x) ))); // G. C. Greubel, Jul 22 2018; Apr 17 2023

A009116 := n->add((-1)^j*binomial(n,2*j),j=0..floor(n/2));

n = 50; (* n = 2 mod 4 *) (CoefficientList[ Series[ Cos[x]/Exp[x], {x, 0, n}], x]* Table[k!, {k,0,n-1}] )[[1 ;; 45]] (* Jean-François Alcover, May 18 2011 *)
Table[(1/2)*((-1-I)^n + (-1+I)^n), {n,0,50}] (* Jean-François Alcover, Jan 31 2018 *)

{a(n) = if( n<0, 0, polcoeff( (1 + x) / (1 + 2*x + 2*x^2) + x * O(x^n), n))} /* Michael Somos, Nov 17 2002 */

def A009116(n): return 2^(n/2)*chebyshev_T(n, -1/sqrt(2))
[A009116(n) for n in range(41)] # G. C. Greubel, Apr 17 2023

Real part of (-1-i)^n. See A009545 for imaginary part. - Marc LeBrun
a(n) = -2 * (a(n-1) + a(n-2)); a(0)=1, a(1)=-1. - Michael Somos, Nov 17 2002
G.f.: (1 + x) / (1 + 2*x + 2*x^2).
E.g.f.: cos(x) / exp(x).
a(n) = Sum_{k=0..n} (-1)^k*A098158(n,k). - Philippe Deléham, Dec 04 2006
a(n)*(-1)^n = A099087(n) - A099087(n-1). - R. J. Mathar, Nov 18 2007
a(n) = (-1)^n*A146559(n). - Philippe Deléham, Dec 01 2008
From Paul Curtz, Jul 22 2011: (Start)
a(n) = -4*a(n-4).
a(n) = A016116(n) * A075553(n+6). (End)
E.g.f.: cos(x)/exp(x) = 1 - x/(G(0)+1), where G(k) = 4k+1-x+(x^2)*(4k+1)/((2k+1)*(4k+3)-(x^2)+x*(2k+1)*(4k+3)/( 2k+2-x+x*(2k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2011
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
a(n) = (-1)^n*2^(n/2)*cos(n*Pi/4). - Nordine Fahssi, Dec 18 2013
a(n) = (-1)^floor((n+1)/2)*2^(n-1)*H(n, n mod 2, 1/2) for n >= 3 where H(n, a, b) = hypergeom([a - n/2, b - n/2], [1 - n], 2). - Peter Luschny, Sep 03 2019
a(n) = 2^(n/2)*ChebyshevT(n, -1/sqrt(2)). - G. C. Greubel, Apr 17 2023
a(n) = A108520(n-1)+A108520(n). - R. J. Mathar, May 09 2023

Extended with signs by Olivier Gérard, Mar 15 1997

Definition corrected by Joerg Arndt, Apr 29 2011

A117575 Expansion of (1-x^3)/((1-x)(1+2x^2)).

Original entry on oeis.org

1, 1, -1, -2, 2, 4, -4, -8, 8, 16, -16, -32, 32, 64, -64, -128, 128, 256, -256, -512, 512, 1024, -1024, -2048, 2048, 4096, -4096, -8192, 8192, 16384, -16384, -32768, 32768, 65536, -65536, -131072, 131072, 262144, -262144, -524288, 524288

Offset: 0

Paul Barry, Mar 29 2006

Row sums of A116949.
From Paul Curtz, Oct 24 2012: (Start)
b(n) = abs(a(n)) = A158780(n+1) = 1,1,1,2,2,4,4,8,8,8,... .
Consider the autosequence (that is a sequence whose inverse binomial transform is equal to the signed sequence) of the first kind of the example. Its numerator is A046978(n), its denominator is b(n). The numerator of the first column is A075553(n).
The denominator corresponding to the 0's is a choice.
The classical denominator is 1,1,1,2,1,4,4,8,1,16,16,32,1,... . (End)

			   0/1,  1/1    1/1,   1/2,   0/2,  -1/4,  -1/4,  -1/8, ...
   1/1,  0/1,  -1/2,  -1/2,  -1/4,   0/4,   1/8,   1/8, ...
  -1/1, -1/2,   0/2,   1/4,   1/4,   1/8,   0/8, -1/16, ...
   1/2,  1/2,   1/4,   0/4   -1/8,  -1/8, -1/16,  0/16, ...
   0/2, -1/4,  -1/4,  -1/8,   0/8,  1/16,  1/16,  1/32, ...
  -1/4,  0/4,   1/8,   1/8,  1/16,  0/16, -1/32, -1/32, ...
   1/4,  1/8,   0/8, -1/16, -1/16, -1/32,  0/32,  1/64, ...
  -1/8, -1/8, -1/16,  0/16,  1/32,  1/32,  1/64,  0/64. - _Paul Curtz_, Oct 24 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,-2).

Cf. A016116, A046978, A075553, A116949, A122016, A191754/A192366.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

[1] cat [(-1)^Floor(n/2)*2^Floor((n-1)/2): n in [1..50]]; // G. C. Greubel, Apr 19 2023

CoefficientList[Series[(1-x^3)/((1-x)(1+2x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{0,-2},{1,1,-1},45] (* Harvey P. Dale, Apr 09 2018 *)

a(n)=if(n,(-1)^(n\2)<<((n-1)\2),1) \\ Charles R Greathouse IV, Jan 31 2012

def A117575(n): return 1 if (n==0) else (-1)^(n//2)*2^((n-1)//2)
[A117575(n) for n in range(51)] # G. C. Greubel, Apr 19 2023

a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) for n >= 3.
a(n) = (cos(Pi*n/2) + sin(Pi*n/2)) * (2^((n-1)/2)*(1-(-1)^n)/2 + 2^((n-2)/2)*(1+(-1)^n)/2 + 0^n/2).
a(n+1) = Sum_{k=0..n} A122016(n,k)*(-1)^k. - Philippe Deléham, Jan 31 2012
E.g.f.: (1 + cos(sqrt(2)*x) + sqrt(2)*sin(sqrt(2)*x))/2. - Stefano Spezia, Feb 05 2023
a(n) = (-1)^floor(n/2)*2^floor((n-1)/2), with a(0) = 1. - G. C. Greubel, Apr 19 2023

A108520 Expansion of 1/(1+2x+2x^2).

Original entry on oeis.org

1, -2, 2, 0, -4, 8, -8, 0, 16, -32, 32, 0, -64, 128, -128, 0, 256, -512, 512, 0, -1024, 2048, -2048, 0, 4096, -8192, 8192, 0, -16384, 32768, -32768, 0, 65536, -131072, 131072, 0, -262144, 524288, -524288, 0, 1048576, -2097152, 2097152, 0, -4194304, 8388608, -8388608

Offset: 0

Michael Somos, Jun 07 2005

Yet another variation on A009545.

Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - R. J. Mathar, Aug 10 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Maran van Heesch, The multiplicative complexity of symmetric functions over a field with characteristic p, Thesis, 2014.
Index entries for linear recurrences with constant coefficients, signature (-2,-2).

a(n) = (-1)^n * A099087(n). a(n) = -A084102(n) if n>0.

Cf. A009116, A009545, A016116, A075553, A084102, A099087.

[n le 2 select n*(-1)^(n-1) else -2*(Self(n-1)+Self(n-2)): n in [1..47]];  // Bruno Berselli, Apr 26 2011

A108520 := n -> `if`(n=0, 1, (-2)^n*hypergeom([1/2-n/2, -n/2], [-n], 2)):
seq(simplify(A108520(n)), n=0..46); # Peter Luschny, Dec 17 2015

CoefficientList[Series[1/(1+2x+2x^2), {x,0,50}], x] (* or *) LinearRecurrence[{-2,-2}, {1,-2}, 50] (* Harvey P. Dale, Sep 30 2012 *)
Table[-(-1-I)^(n-1) - (-1+I)^(n-1), {n, 0, 50}] (* Bruno Berselli, Nov 08 2015 *)
Im[(-1+I)^Range[51]] (* G. C. Greubel, Apr 24 2023 *)

a(n)=if(n<0, 0, polcoeff(1/(1+2*x+2*x^2)+x*O(x^n),n))

a(n)=if(n<1, n==0, -polsym(2+2*x+x^2,n-1)[n])

vector(66,n,imag((-1+I)^n)) /* Joerg Arndt, May 13 2011 */

[imag((-1+I)^(n+1)) for n in range(51)] # G. C. Greubel, Apr 24 2023

G.f.: 1/(1+2*x+2*x^2).
E.g.f.: exp(-x)*(cos(x) - sin(x)).
a(n) = -2*(a(n-1) + a(n-2)).
a(n) = Sum_{k=0..n} Sum_{j=0..n-k} C(k,j)*C(k,n-j)*(-2)^(n-j). - Paul Barry, Mar 09 2006
a(n) = -4 * a(n-4). - Paul Curtz, Apr 24 2011
a(n) = A016116(n+1) * A075553(n+1). - Paul Curtz, Apr 25 2011
From Bruno Berselli, Apr 26 2011: (Start)
a(n) = -(-1-i)^(n-1) - (-1+i)^(n-1), where i=sqrt(-1).
a(n) = -2*A009116(n-1) for n > 0. (End)
Imaginary part of (-1+i)^n, negated real part is A090132. - Joerg Arndt, May 13 2011
E.g.f.: (cos(x) - sin(x))*exp(-x) = G(0); G(k) = 1 - 2*x/(4*k+1+x*(4*k+1)/(2*(2*k+1) -x -2*(x^2)*(2*k+1)/((x^2) -(2*k+2)*(4*k+3)/G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2011
G.f.: G(0)/(2*(1+x)), where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+2) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 20 2013
a(n) = (-2)^n*hypergeom([1/2-n/2, -n/2], [-n], 2) for n >= 1. - Peter Luschny, Dec 17 2015

A075554 Denominators in the Maclaurin series for arctan(1+x).

Original entry on oeis.org

2, 4, 12, 1, 40, 48, 112, 1, 288, 320, 704, 1, 1664, 1792, 3840, 1, 8704, 9216, 19456, 1, 43008, 45056, 94208, 1, 204800, 212992, 442368, 1, 950272, 983040, 2031616, 1, 4325376, 4456448, 9175040, 1, 19398656, 19922944, 40894464, 1, 85983232

Offset: 1

Eric W. Weisstein, Sep 23 2002

Terms with mod(n,4)=0 are zero, so a(n)=1 for those n.

arctan(1 + x) = Pi/4 + integral_{0..x} dt / (2 + 2*t + t^2). - Michael Somos, Apr 20 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A075553, A132314.

Table[Denominator[(-1)^n*2^(-n-1)*((1+I)^n-(1-I)^n)*I/n], {n, 1, 41}] (* Jean-François Alcover, Apr 18 2014, after Vladimir Kruchinin *)

atan(n):=(sum((sum((2^(i-n-1)*(-1)^(i+n+(k-1)/2)/i*binomial(k-1,k-i)),i,1,k))*binomial(n-1,n-k),k,1,n));
makelist(denom(atan(n),n,1,10); /* Vladimir Kruchinin, Apr 17 2014 */

a(n) = Denominator(sum(k=1..n, (sum(i=1..k, (2^(i-n-1)*(-1)^(i+n+(k-1)/2)/i*binomial(k-1,k-i))))*binomial(n-1,n-k))). - Vladimir Kruchinin, Apr 17 2014

Empirical g.f.: -x*(16*x^11 -16*x^10 -16*x^9 -24*x^8 -8*x^7 +4*x^6 +12*x^5 +22*x^4 +x^3 +12*x^2 +4*x +2) / ((x -1)*(x +1)*(x^2 +1)*(2*x^2 -1)^2*(2*x^2 +1)^2). - Colin Barker, Apr 18 2014

A339513 Define R_{1}(x)=1, R_{n+1}(x)=(R_n(x)2x/(1+x^2))'; then a(n)=R_{n}(1).

Original entry on oeis.org

1, 0, -1, 3, -2, -45, 347, -756, -13031, 184245, -810034, -11404503, 264733177, -1931955480, -21453955777, 796153961091, -8688345850874, -69492467459925, 4300450718587619, -65896562313762012, -307002797419794407, 37668399518087366325

Offset: 1

Luc Rousseau, Dec 07 2020

sign

Let (R_n) be the sequence of rational functions satisfying: R_1(x) = 1; R_{n+1}(x) = (R_n(x) * 2*x/(1+x^2))'. By definition, a(n) = R_n(1).

Applying [Dominici, Theorem 4.1] proves that the e.g.f. of this sequence is the series reversion of log(1+x)/2 + x^2/4 + x/2.

			R_1(x) = 1,
  so a(1) = R_1(1) = 1.
R_2(x) = (R_1(x)*2*x/(1+x^2))' = (1 * 2*x/(1+x^2))' = 2*(1-x^2)/(1+x^2)^2,
  so a(2) = R_2(1) = 0.
R_3(x) = (R_2(x)*2*x/(1+x^2))' = (2*(1-x^2)/(1+x^2)^2 * 2*x/(1+x^2))' = 4*(1-8*x^2+3*x^4)/(1+x^2)^4, so a(3) = R_3(1) = -1.

Luc Rousseau, Table of n, a(n) for n = 1..300
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.

Cf. A214406, A145271, A090932, A075553.

list_a(nmax)=my(n,r);n=1;r=1;print1(subst(r,x,1),", ");while(n

my(x='x+O('x^33)); Vec(serlaplace(serreverse(log(1+x)/2 + x^2/4 + x/2))) \\ Joerg Arndt, Dec 22 2020

a(n) = (Sum_{k=0..n-1} (-1)^k*A214406(n-1,k))/2^(n-1).
a(n) = Sum_{P partition of n-1} A145271(P) * Product_{p part of P} A090932(p)*A075553(p+3).
E.g.f.: series reversion of log(1+x)/2 + x^2/4 + x/2.

cp's OEIS Frontend

A166486 Periodic sequence [0,1,1,1] of length 4; Characteristic function of numbers that are not multiples of 4.

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A009116 Expansion of e.g.f. cos(x) / exp(x).

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A117575 Expansion of (1-x^3)/((1-x)*(1+2*x^2)).

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A108520 Expansion of 1/(1+2*x+2*x^2).

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A075554 Denominators in the Maclaurin series for arctan(1+x).

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A339513 Define R_{1}(x)=1, R_{n+1}(x)=(R_n(x)*2*x/(1+x^2))'; then a(n)=R_{n}(1).

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A117575 Expansion of (1-x^3)/((1-x)(1+2x^2)).

A108520 Expansion of 1/(1+2x+2x^2).

A339513 Define R_{1}(x)=1, R_{n+1}(x)=(R_n(x)2x/(1+x^2))'; then a(n)=R_{n}(1).