A087204 -id:A087204 - cp's OEIS frontend

A138230 Expansion of (1-x)/(1 - 2x + 4x^2).

1, 1, -2, -8, -8, 16, 64, 64, -128, -512, -512, 1024, 4096, 4096, -8192, -32768, -32768, 65536, 262144, 262144, -524288, -2097152, -2097152, 4194304, 16777216, 16777216, -33554432, -134217728, -134217728, 268435456, 1073741824, 1073741824, -2147483648, -8589934592

Offset: 0

Paul Barry, Mar 06 2008

In general, the expansion of (1-x)/(1 - 2*x + (m+1)*x^2) has general term given by a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*(-m)^k = ((1+sqrt(-m))^n + (1-sqrt(-m))^n)/2.

Binomial transform of [1, 0, -3, 0, 9, 0, -27, 0, 81, 0, ...] = powers of -3 with interpolated zeros. - Philippe Deléham, Dec 02 2008

Michael De Vlieger, Table of n, a(n) for n = 0..3322
Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras, Vol. 29 (2019), Article 54.
Index entries for linear recurrences with constant coefficients, signature (2,-4).

Cf. A087204, A088138, A098158, A104537, A124182, A128018.

[2^n*Evaluate(ChebyshevFirst(n), 1/2): n in [0..30]]; // G. C. Greubel, Feb 11 2023

CoefficientList[Series[(1-x)/(1-2x+4x^2),{x,0,30}],x] (* or *) LinearRecurrence[{2,-4},{1,1},30] (* Harvey P. Dale, Nov 11 2014 *)

[2^n*chebyshev_T(n,1/2) for n in range(31)] # G. C. Greubel, Feb 11 2023

From Philippe Deléham, Nov 14 2008: (Start)
a(n) = 2*a(n-1) - 4*a(n-2), a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A098158(n,k)*(-3)^(n-k). (End)
a(n) = Sum_{k=0..n} A124182(n,k)*(-4)^(n-k). - Philippe Deléham, Nov 15 2008
a(n) = 2^n*cos(Pi*n/3). - Richard Choulet, Nov 19 2008
a(n) = -8*a(n-3). - Paul Curtz, Apr 22 2011
From Sergei N. Gladkovskii, Jul 27 2012: (Start)
G.f.: G(0) where G(k) = 1 + x/(1 + 2*x/(1 - 2*x - 4*x/(4*x + 1/G(k+1)))); (continued fraction).
E.g.f.: exp(x)*cos(sqrt(3)*x) = G(0) where G(k) = 1 + x/(3*k+1 + 2*x*(3*k+1)/(3*k+2 - 2*x - 4*x*(3*k+2)/(4*x + 3*(k+1)/G(k+1)))); (continued fraction). (End)
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(3*k+1)/(x*(3*k+4) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = A088138(n+1) - A088138(n). - R. J. Mathar, Mar 04 2018
a(n) = (-1)^n*A104537(n). - R. J. Mathar, May 21 2019
a(n) = 2^(n-1)*A087204(n). - G. C. Greubel, Feb 11 2023
Sum_{n>=0} 1/a(n) = 4/3. - Amiram Eldar, Feb 14 2023

A100051 A Chebyshev transform of 1,1,1,...

Original entry on oeis.org

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

Offset: 0

Paul Barry, Oct 31 2004

1, followed by period 6: repeat [1, -1, -2, -1, 1, 2]. - Joerg Arndt, Aug 28 2024
A Chebyshev transform of 1/(1-x): if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).
Transform of 1/(1+x) under the mapping g(x)->((1+x)/(1-x))g(x/(1-x)^2). - Paul Barry, Dec 01 2004
Multiplicative with a(p^e) = -1 if p = 2; -2 if p = 3; 1 otherwise. - David W. Wilson, Jun 10 2005

			G.f. = 1 + x - x^2 - 2*x^3 - x^4 + x^5 + 2*x^6 + x^7 - x^8 - 2*x^9 - x^10 + ...

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).

Cf. A011655, A057079, A087204, A099837, A099443, A100047, A100048, A100050.

Row sums of array A127677.

CoefficientList[Series[(1 - x^2)/(1 - x + x^2), {x,0,50}], x] (* G. C. Greubel, May 03 2017 *)
LinearRecurrence[{1,-1},{1,1,-1},80] (* Harvey P. Dale, Mar 25 2019 *)

{a(n) = - (n == 0) + [2, 1, -1, -2, -1, 1][n%6 + 1]}; /* Michael Somos, Mar 21 2011 */

From Paul Barry, Dec 01 2004: (Start)
G.f.: (1-x^2)/(1-x+x^2).
a(n) = a(n-1) - a(n-2), n>2.
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)/(n-k).
a(n) = Sum_{k=0..n} binomial(n+k, 2k)*(2n/(n+k))*(-1)^k, n>1. (End)
Moebius transform is length 6 sequence [1, -2, -3, 0, 0, 6].
Euler transform of length 6 sequence [1, -2, -1, 0, 0, 1].
a(n) = a(-n). a(n) = c_6(n) if n>1 where c_k(n) is Ramanujan's sum. - Michael Somos, Mar 21 2011
a(n) = A087204(n), n>0. - R. J. Mathar, Sep 02 2008
a(n) = A057079(n+1), n>0. Dirichlet g.f. zeta(s) *(1-2^(1-s)-3^(1-s)+6^(1-s)). - R. J. Mathar, Apr 11 2011

A131027 Period 6: repeat [4, 3, 1, 0, 1, 3].

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, following a suggestion of Paul Curtz, Jun 10 2007

Third column of triangular array T defined in A131022.
a(n) = abs(A078070(n+1)).
Determinants of the spiral knots S(3,k,(1,1)). a(k+4) = det(S(3,k,(1,1))). These knots are also the torus knots T(3,k). - Ryan Stees, Dec 13 2014

			For k=3, b(7)=sqrt(3)b(6)-b(5)=3-1=2, so det(S(3,3,(1,1)))=2^2=4.

A. Breiland, L. Oesper, and L. Taalman, p-Coloring classes of torus knots, Online Missouri J. Math. Sci., 21 (2009), 120-126.
N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, Spiral knots, Missouri J. of Math. Sci., 22 (2010).
M. DeLong, M. Russell, and J. Schrock, Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m), Involve, Vol. 8 (2015), No. 3, 361-384.
Seong Ju Kim, R. Stees, and L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
Index entries for linear recurrences with constant coefficients, signature (2,-2,1).

Cf. A087204, A131022, A078070. Other columns of T are in A088911, A131026, A131028, A131029, A131030.

m:=105; [ [4, 3, 1, 0, 1, 3][(n-1) mod 6 + 1]: n in [1..m] ];

A131027:=n->2+cos(n*Pi/3)+sqrt(3)*sin(n*Pi/3): seq(A131027(n), n=1..100); # Wesley Ivan Hurt, Sep 11 2014

Table[2 + Cos[n*Pi/3] + Sqrt[3]*Sin[n*Pi/3], {n, 30}] (* Wesley Ivan Hurt, Sep 11 2014 *)

{m=105; for(n=1, m, r=(n-1)%6; print1(if(r==0, 4, if(r==1||r==5, 3, if(r==3, 0, 1))), ","))}

[(lucas_number2(n,2,1)-lucas_number2(n-1,1,1)) for n in range(4, 109)] # Zerinvary Lajos, Nov 10 2009

a(1) = 4, a(2) = a(6) = 3, a(3) = a(5) = 1, a(4) = 0, a(6) = 1; for n > 6, a(n) = a(n-6).
G.f.: (4-5*x+3*x^2)/((1-x)*(1-x+x^2)).
a(n) = 2+cos(n*Pi/3)+sqrt(3)*sin(n*Pi/3) = 2+(-1)^((n-1)/3)+(-1)^((1-n)/3). - Wesley Ivan Hurt, Sep 11 2014
a(k+4) = det(S(3,k,(1,1))) = (b(k+4))^2, where b(5)=1, b(6)=sqrt(3), b(k)=sqrt(3)*b(k-1) - b(k-2) = b(6)*b(k-1) - b(k-2). - Ryan Stees, Dec 13 2014
a(n) = 2 + 2*cos(Pi/3*(n-1)) = 2 + A087204(n-1) for n >= 1. - Werner Schulte, Jul 18 2017 and Peter Munn, Apr 28 2022

A111927 Expansion of x^3 / ((x-1)(2x-1)*(x^2-x+1)).

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 21, 42, 84, 169, 340, 682, 1365, 2730, 5460, 10921, 21844, 43690, 87381, 174762, 349524, 699049, 1398100, 2796202, 5592405, 11184810, 22369620, 44739241, 89478484, 178956970, 357913941, 715827882, 1431655764, 2863311529, 5726623060

Offset: 0

Creighton Dement, Aug 21 2005

Binomial transform of sequence (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0). Note: the binomial transform of the sequence (0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0) is A111926; the binomial transform of the sequence (0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0) is A024495 (disregarding first two terms, which are both zero).

The sequence relates the calculation of the logarithm of the Twin Prime Constants of order 3 to the sequence of prime zeta functions, see definition 7 in arXiv:0903.2514. - R. J. Mathar, Mar 28 2009

Colin Barker, Table of n, a(n) for n = 0..1000
Antoine-Augustin Cournot, Solution d'un problème d'analyse combinatoire, Bulletin des Sciences Mathématiques, Physiques et Chimiques, item 34, volume 11, 1829, pages 93-97. Also at Google Books. Page 97 case p=3 formula y^(0) = a(n). (But misprint "- (2/3)*cos" should be "+ (2/3)*cos".)
R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011.
Christian Ramus, Solution générale d'un problème d'analyse combinatoire, Journal für die Reine und Angewandte Mathematik (Crelle's journal), volume 11, 1834, pages 353-355. Page 353 case p=3 formula y^(0) = a(n). (But misprint "+ (1/3)*cos" should be "+ (2/3)*cos", per the general case equation A page 354.)
Kevin Ryde, Iterations of the Terdragon Curve, section Lines, quantity Lines_k(2) = a(k+1).
Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-2).

Cf. A000295, A111926, A024495, A131708.

seq(sum(binomial(n, k*3), k=1..n), n=0..33); # Zerinvary Lajos, Oct 23 2007

LinearRecurrence[{4,-6,5,-2},{0,0,0,1},40] (* Harvey P. Dale, Jul 04 2017 *)

concat(vector(3), Vec(x^3/((x-1)*(2*x-1)*(x^2-x+1)) + O(x^40))) \\ Colin Barker, Feb 10 2017

a(n+2) - a(n+1) + a(n) = A000225(n).
a(n) - a(n-1) = A024495(n-1).
From Colin Barker, Feb 10 2017: (Start)
a(n) = 2^n/3 + 2*cos((Pi*n)/3)/3 - 1. [Cournot]
a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 2*a(n-4) for n > 3. (End)
a(n) = (2^n+A087204(n))/3 - 1. - R. J. Mathar, Aug 07 2017
a(n) = (1/3)*Sum_{k=0..n-1} binomial(n, 3*floor(k/3)+3). - Taras Goy, Jan 26 2025
E.g.f.: (exp(x)*(exp(x) - 3) + 2*exp(x/2)*cos(sqrt(3)*x/2))/3. - Stefano Spezia, Feb 06 2025

A100886 Expansion of x(1+3x+2x^2)/((1+x+x^2)(1-x-x^2)).

Original entry on oeis.org

0, 1, 3, 3, 5, 10, 14, 23, 39, 61, 99, 162, 260, 421, 683, 1103, 1785, 2890, 4674, 7563, 12239, 19801, 32039, 51842, 83880, 135721, 219603, 355323, 574925, 930250, 1505174, 2435423, 3940599, 6376021, 10316619, 16692642, 27009260, 43701901

Offset: 0

Creighton Dement, Nov 21 2004

This sequence was investigated in cooperation with Paul Barry.
Generating floretion: - 0.5'i - 0.5'k - 0.5j' - 0.5'ii' + 0.5'jj' - 0.5'kk' + 0.5'ik' - 0.5'ki' ("tes").
From Joshua P. Bowman, Sep 28 2023: (Start)
a(n) is equal to the number of circular binary sequences of length n+1 with an even number of 0's and no consecutive 1's. A circular binary sequence is a finite sequence of 0's and 1's for which the first and last digits are considered to be adjacent. Rotations are distinguished from each other.
a(n) is also equal to the number of matchings in the cycle graph C_{n+1} for which the number of edges plus the number of unmatched vertices is even.
a(n) is also equal to the number of circular compositions of n+1 into an even number of 1's and 2's. (End)

			When counting circular binary sequences with an even number of 0's and no consecutive 1's, the sequence "1" is not allowed because the 1 is considered to be adjacent to itself. Thus a(0)=0. For n=2, the a(2)=3 allowed sequences of length 3 are 001, 010, and 100. For n=3, the a(3)=3 allowed sequences of length 4 are 0000, 0101, and 1010. - _Joshua P. Bowman_, Sep 28 2023

Wojciech Florek, Table of n, a(n) for n = 0..1001
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 19.
Wojciech Florek, A class of generalized Tribonacci sequences applied to counting problems, Appl. Math. Comput., 338 (2018), 809-821.
W. O. J. Moser, Cyclic binary strings without long runs of like (alternating) bits, Fibonacci Quart. 31 (1993), no. 1, 2-6.
Index entries for linear recurrences with constant coefficients, signature (0,1,2,1).

Cf. A000032, A007040, A087204, A100887, A100888, A100889, A100890, A366043.

I:=[0,1,3,3]; [n le 4 select I[n] else Self(n-2)+2*Self(n-3)+Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 30 2015

a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 3; a[n_] := a[n] = a[n - 2] + 2a[n - 3] + a[n - 4]; Table[ a[n], {n, 0, 36}]
(* Or *) CoefficientList[ Series[x(1 + 3x + 2x^2)/((1 + x + x^2)(1 - x - x^2)), {x, 0, 36}], x] (* Robert G. Wilson v, Nov 26 2004 *)
LinearRecurrence[{0,1,2,1},{0,1,3,3},40] (* Harvey P. Dale, Apr 04 2016 *)

a(n):=n*sum(binomial(k,n-k)*(if oddp(k) then 0 else 1/k),k,1,n); /* Vladimir Kruchinin, Apr 09 2011 */

a(n)=n*sum(j=1,n\2,k=2*j;binomial(k,n-k)/k);
vector(66,n,a(n)) /* Joerg Arndt, Apr 09 2011 */

concat([0],Vec(x*(1+3*x+2*x^2)/((1+x+x^2)*(1-x-x^2))+O(x^66))) /* Joerg Arndt, Apr 09 2011 */

(1/2)*(a(n) + A100887(n) - A100888(n)) = A061347(n+3).
a(n) = (L(n+1)-A061347(n+1))/2, L=A000032; [corrected by Wojciech Florek, Feb 26 2018]
a(n) = a(n-2) + 2*a(n-3) + a(n-4), a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 3.
a(n) = n*Sum_{j=1..floor(n/2)} binomial(2*j,n-2*j)/(2*j). - Vladimir Kruchinin, Apr 09 2011 (with offset 1, cf. PARI code)
a(n) = floor(phi^(n+1)/2), n mod 3 = 0,1; a(n) = floor((phi^(n+1)+3)/2), n mod 3 = 2, phi = (1 + sqrt(5))/2; from Binet's formula or the relation to the Lucas numbers A000032. - Wojciech Florek, Mar 03 2018
a(n) = A000032(n+1) - A366043(n+1). - Joshua P. Bowman, Sep 28 2023

More terms from Robert G. Wilson v, Nov 26 2004

A130781 Sequence is identical to its third differences: a(n+3) = 3a(n+2) - 3a(n+1) + 2*a(n), with a(0)=a(1)=1, a(2)=2.

Original entry on oeis.org

1, 1, 2, 5, 11, 22, 43, 85, 170, 341, 683, 1366, 2731, 5461, 10922, 21845, 43691, 87382, 174763, 349525, 699050, 1398101, 2796203, 5592406, 11184811, 22369621, 44739242, 89478485, 178956971, 357913942, 715827883, 1431655765, 2863311530

Offset: 0

Paul Curtz, Jul 14 2007, Jul 18 2007

The inverse binomial transform is 1,0,1,... repeated with period 3, essentially A011655. - R. J. Mathar, Aug 28 2023

Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

See A130750, A130752, A130755, A129339.

Essentially a duplicate of A024493.

a[n_] := a[n] = 3 a[n - 1] - 3 a[n - 2] + 2 a[n - 3]; a[0] = a[1] = 1; a[2] = 2; Table[a@n, {n, 0, 33}] (* Or *)
CoefficientList[ Series[(1 - 2 x + 2 x^2)/(1 - 3 x + 3 x^2 - 2 x^3), {x, 0, 33}], x] (* Robert G. Wilson v, Sep 08 2007 *)
LinearRecurrence[{3,-3,2},{1,1,2},40] (* Harvey P. Dale, Sep 17 2013 *)

3*a(n) = 2^(n+1) + A087204(n+1).
Also first differences of A024494.
G.f.: (1-2x+2x^2)/(1-3x+3x^2-2x^3).
Binomial transform of [1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, ...]; i.e., ones in positions 2, 5, 8, 11, ... and the rest zeros. [Corrected by Gary W. Adamson, Jan 07 2008]

Edited by N. J. A. Sloane, Jul 28 2007

A087205 a(n) = -2a(n-1) + 4a(n-2), a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 0, 8, -16, 64, -192, 640, -2048, 6656, -21504, 69632, -225280, 729088, -2359296, 7634944, -24707072, 79953920, -258736128, 837287936, -2709520384, 8768192512, -28374466560, 91821703168, -297141272576, 961569357824, -3111703805952

Offset: 0

Paul Barry, Aug 25 2003

Inverse binomial transform of A087204.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,4)

Cf. A000045, A063727, A084222, A085449, A087204.

[(-1)^(n+1)*2^n*Fibonacci(n-2): n in [0..50]]; // G. C. Greubel, Oct 08 2018

Table[-(-2)^n*Fibonacci[n - 2], {n, 0, 50}] (* G. C. Greubel, Oct 08 2018 *)
LinearRecurrence[{-2,4},{1,2},30] (* Harvey P. Dale, Jan 24 2022 *)

Vec((4*x+1)/(-4*x^2+2*x+1)+O(x^66)) \\ Joerg Arndt, Jul 14 2013

vector(50, n, n--; (-1)^(n+1)*2^n*fibonacci(n-2)) \\ G. C. Greubel, Oct 08 2018

a(n) = (-1-sqrt(5))^n * (1/2-3*sqrt(5)/10) + (-1+sqrt(5))^n * (1/2+3*sqrt(5)/10).
G.f.: (4*x +1)/(-4*x^2 +2*x +1). - Joerg Arndt, Jul 14 2013
a(n+2) = A085449(n)*(-1)^(n+1); a(n+3) = A063727(n)*(-1)^n.
a(n) = -(-2)^n*F(n-2) for n >= 0, with F = A000045, and F(-1) = 1, F(-2) = -1. - Wolfdieter Lang, Oct 08 2018

A353446 Let g be the inverse Möbius transform of the Eisenstein integer-valued function f defined in A353445. a(n) is twice the real part of g(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen and Peter Munn, Apr 19 2022

sign

The imaginary part of g(n) is A353354(n)*(sqrt(3)/2)*i.
f(n), g(n), and so also a(n), are determined by the cubefree part of n, A050985(n). If the cubefree part is not squarefree, g(n) is 0; otherwise g(n) = x^(A195017(A050985(n))), where x = (1 + sqrt(3)*i)/2, the primitive 6th root of unity with positive imaginary part.
The above formula arises from g being multiplicative (because f is multiplicative). g(prime(m)^k) is 1 for k == 0 (mod 3), 0 for k == 2 (mod 3), and for k == 1 (mod 3) the result depends on the parity of m. g(prime(m)^(3k+1)) is 1+w for odd m, -w for even m, where w is the cube root of unity with positive imaginary part. 1+w and -w are the primitive 6th roots of unity.
So the range of g is the 6 sixth roots of unity and 0 itself: these are the 7 Eisenstein integers closest to 0, and they are clearly closed under multiplication. The range of (a(n)) is [-2..2]. g(n) and a(n) are 0 if and only if the cubefree part of n is not squarefree. (Compare with the Moebius function being 0 when its argument is not squarefree.) Otherwise a(n) is even if and only if n is in A332820.

Peter Munn, Table of n, a(n) for n = 1..10000
Peter Munn, Figure showing relationship of the Eisenstein integer g(n) to the presence of n in other sequences.
Eric Weisstein's World of Mathematics, Eisenstein Integer

Sequences used in a definition of this sequence: A008966, A050985, A090882, A087204, A195017, A353445.
See A353354 for the imaginary part.
Positions of particular values depend on A059269, A211337, A211338, A332820 as shown in the formula section.
Positions of even numbers: A353355.

A050985(n) = { my(f=factor(n)); f[, 2] = apply(x->(x % 3), f[, 2]); factorback(f); }; \\ From A050985
A087204(n) = ([2, 1, -1, -2, -1, 1][1+(n%6)]);
A195017(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * (-1)^(1+primepi(f[i,1])))); };
A353446(n) = { my(u=A050985(n)); issquarefree(u) * A087204(abs(A195017(u))); };

a(n) = A008966(m) * A087204(A090882(m)) = A008966(m) * A087204(|A195017(m)|), where m = A050985(n), the cubefree part of n, and A008966(.) is the characteristic function of squarefree numbers.
For n >= 1, -2 <= a(n) <= 2.
{n : a(n) = -2} = {A211338} INTERSECT {A332820}.
{n : a(n) = -1} = {A211337} \ {A332820}.
{n : a(n) = 0} = {A059269}.
{n : a(n) = 1} = {A211338} \ {A332820}.
{n : a(n) = 2} = {A211337} INTERSECT {A332820}.

A272931 a(n) = 2^(n+1)cos(narctan(sqrt(15))).

Original entry on oeis.org

2, 1, -7, -11, 17, 61, -7, -251, -223, 781, 1673, -1451, -8143, -2339, 30233, 39589, -81343, -239699, 85673, 1044469, 701777, -3476099, -6283207, 7621189, 32754017, 2269261, -128746807, -137823851, 377163377, 928458781, -580194727, -4294029851, -1973250943

Offset: 0

Peter Luschny, May 11 2016

For n >= 1, |a(n)| is the unique odd positive solution y to 4^(n+1) = 15*x^2 + y^2. The value of x is |A106853(n-1)|. - Jianing Song, Jan 22 2019

Colin Barker, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,-4).

Cf. A087204, A002249, A106853.

seq(simplify(((1-I*sqrt(15))^n + (1+I*sqrt(15))^n)/2^n), n=0..32);

Mathematica
```
LinearRecurrence[{1, -4}, {2, 1}, 33]
```

Vec((2 - x) / (1 - x + 4*x^2) + O(x^40)) \\ Colin Barker, Jan 22 2019

[lucas_number2(i, 1, 4) for i in range(33)]

Let a(x) = x/2 - i*sqrt(15)*x/2 and b(x) = x/2 + i*sqrt(15)*x/2, then:
a(n) = a(1)^n + b(1)^n.
a(n) = n! [x^n] exp(a(x)) + exp(b(x)).
a(n) = [x^n] (2 - x)/(4*x^2 - x + 1).
a(n) = Sum_{k=0..floor(n/2)} (-4)^k*n*(n - k - 1)!/(k!*(n - 2*k)!) for n >= 1.
For n >= 1, 15*a(n)^2 + A106853(n-1)^2 = 4^(n+1). - Jianing Song, Jan 22 2019
a(n) = a(n-1) - 4*a(n-2) for n>1. - Colin Barker, Jan 22 2019
a(n) = 2*A106853(n) - A106853(n-1). - R. J. Mathar, Aug 19 2022

A122876 a(0)=1, a(1)=1, a(2)=2, a(n) = a(n-1) - a(n-2) for n>2.

Original entry on oeis.org

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

Offset: 0

Philippe Deléham, Oct 24 2006

Essentially the same as A057079, A087204 and A100051.

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

Cf. A055830, A057079, A087204, A100051.

LinearRecurrence[{1,-1}, {1,1,2}, 50] (* G. C. Greubel, May 03 2017; corrected by Georg Fischer, Apr 02 2019 *)
(* or *) CoefficientList[Series[(1 + 2*x^2)/(1 - x + x^2), {x,0,50}], x] (* G. C. Greubel, May 03 2017 *)

my(x='x+O('x^50)); Vec((1+2*x^2)/(1-x+x^2)) \\ G. C. Greubel, May 03 2017

a(n) = Sum_{k=0..floor(n/2)} (-2)^k*A055830(n-k,k).

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

cp's OEIS Frontend

A138230 Expansion of (1-x)/(1 - 2*x + 4*x^2).

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A100051 A Chebyshev transform of 1,1,1,...

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A131027 Period 6: repeat [4, 3, 1, 0, 1, 3].

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A111927 Expansion of x^3 / ((x-1)*(2*x-1)*(x^2-x+1)).

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A100886 Expansion of x*(1+3*x+2*x^2)/((1+x+x^2)*(1-x-x^2)).

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A130781 Sequence is identical to its third differences: a(n+3) = 3*a(n+2) - 3*a(n+1) + 2*a(n), with a(0)=a(1)=1, a(2)=2.

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A087205 a(n) = -2*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.

A138230 Expansion of (1-x)/(1 - 2x + 4x^2).

A111927 Expansion of x^3 / ((x-1)(2x-1)*(x^2-x+1)).

A100886 Expansion of x(1+3x+2x^2)/((1+x+x^2)(1-x-x^2)).

A130781 Sequence is identical to its third differences: a(n+3) = 3a(n+2) - 3a(n+1) + 2*a(n), with a(0)=a(1)=1, a(2)=2.

A087205 a(n) = -2a(n-1) + 4a(n-2), a(0)=1, a(1)=2.

A272931 a(n) = 2^(n+1)cos(narctan(sqrt(15))).