A105309 -id:A105309 - cp's OEIS frontend

A218134 Norm of coefficients in the expansion of 1/(1 - 2x - ix^2), where i is the imaginary unit.

1, 4, 17, 80, 369, 1700, 7841, 36160, 166753, 768996, 3546289, 16354000, 75417809, 347795396, 1603886913, 7396455680, 34109360321, 157298104900, 725393076049, 3345209499600, 15426707209777, 71141522037604, 328074947492321, 1512944453384000, 6977067089461281

Offset: 0

Paul D. Hanna, Oct 21 2012

nonn

The radius of convergence of g.f. equals 1 + sqrt(2) - sqrt(2)*sqrt(1 + sqrt(2)) = 0.216845335...

The following remarks assume an offset of 1. This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. The sequence satisfies a linear recurrence of order 4. It is the case P1 = 4, P2 = -4, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

			G.f.: A(x) = 1 + 4*x + 17*x^2 + 80*x^3 + 369*x^4 + 1700*x^5 + 7841*x^6 +...
The terms equal the norm of the complex coefficients in the expansion:
1/(1 - 2*x - i*x^2) = 1 + 2*x + (4 + i)*x^2 + (8 + 4*i)*x^3 + (15 + 12*i)*x^4 + (26 + 32*i)*x^5 + (40 + 79*i)*x^6 + (48 + 184*i)*x^7 +...
so that
a(1) = 2^2, a(2) = 4^2 + 1, a(3) = 8^2 + 4^2, a(4) = 15^2 + 12^2, a(5) = 26^2 + 32^2, ...

Peter Bala, Linear divisibility sequences and Chebyshev polynomials
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (4,2,4,-1).

Cf. A105309, A218135.

Cf. also A000129, A057087, A100047.

LinearRecurrence[{4, 2, 4, -1}, {1, 4, 17, 80}, 25] (* Jean-François Alcover, Nov 02 2019 *)

{a(n)=norm(polcoeff(1/(1-2*x-I*x^2+x*O(x^n)), n))}
for(n=0,31,print1(a(n),", "))

G.f.: (1 - x^2)/(1 - 4*x - 2*x^2 - 4*x^3 + x^4).
From Peter Bala, Mar 25 2014: (Start)
The following formulas assume an offset of 1.
a(n) = 1/(2*sqrt(2))*(T(n,1 + sqrt(2)) - T(n,1 - sqrt(2))), where T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 1; 1, 2]. Note, the bottom left element of the matrix M^n gives the Lucas sequence A000129.
a(n) = U(n-1,exp(2*i*Pi/8))*U(n-1,exp(-2*i*Pi/8)) = U(n-1,(1 + i)/sqrt(2))*U(n-1,(1 - i)/sqrt(2)), where U(n,x) denotes the Chebyshev polynomial of the second kind.
The o.g.f. is the Chebyshev transform of the rational function x/(1 - 4*x - 4*x^2), where the Chebyshev transform takes the function A(x) to the function (1 - x^2)/(1 + x^2)*A(x/(1 + x^2)). See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)

A201837 G.f.: real part of 1/(1 - ix - ix^2) where i=sqrt(-1).

Original entry on oeis.org

1, 0, -1, -2, 0, 4, 5, -2, -13, -12, 12, 40, 25, -52, -117, -38, 196, 324, -3, -678, -841, 360, 2200, 2000, -2079, -6760, -4121, 8918, 19720, 6084, -33435, -54442, 1547, 115228, 140772, -63880, -372775, -332892, 359763, 1142322, 678796, -1528956, -3323203

Offset: 0

Paul D. Hanna, Dec 06 2011

The norm of the coefficients in 1/(1 - i*x - i*x^2) is given by A105309.

			G.f.: A(x) = 1 - x^2 - 2*x^3 + 4*x^5 + 5*x^6 - 2*x^7 - 13*x^8 - 12*x^9 +...
A201838 gives the imaginary part of coefficients in 1/(1 -i*x - i*x^2) and begins: 0, 1, 1, -1, -3, -2, 4, 9, 3, -15, -25, 0, 52, 65, -27, -169, -155, 158, 520,... in which this sequence equals the negative of the pairwise sums of A201838.

Paul D. Hanna, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (0,-1,-2,-1).

Cf. A201838 (imag), A105309 (norm).

Re/@ CoefficientList[Series[1/(1-I*x-I*x^2),{x,0,50}],x] (* Harvey P. Dale, Dec 10 2011 *)

{a(n)=real(polcoeff(1/(1-I*x-I*x^2+x*O(x^n)),n))}

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

G.f.: 1/(1 + x^2*(1+x)^2).
a(n) = -(A201838(n-1) + A201838(n-2)), where A201838 gives the imaginary part of the coefficients in 1/(1 - i*x - i*x^2).
a(n) = Re((((i + sqrt(4*i-1))^(n+1) - (i - sqrt(4*i-1))^(n+1)))/(2^(n+1)*sqrt(4*i-1))), where i=sqrt(-1). - Daniel Suteu, Apr 20 2018
a(n) = - a(n-2) - 2*a(n-3) - a(n-4). - Wesley Ivan Hurt, Sep 05 2022

A201838 G.f.: imaginary part of 1/(1 - ix - ix^2) where i=sqrt(-1).

Original entry on oeis.org

0, 1, 1, -1, -3, -2, 4, 9, 3, -15, -25, 0, 52, 65, -27, -169, -155, 158, 520, 321, -681, -1519, -481, 2560, 4200, -79, -8839, -10881, 4797, 28638, 25804, -27351, -87877, -52895, 116775, 256000, 76892, -436655, -705667, 26871, 1502085, 1821118, -850160

Offset: 0

Paul D. Hanna, Dec 06 2011

The norm of the coefficients in 1/(1 - i*x - i*x^2) is given by A105309.

			G.f.: A(x) = x + x^2 - x^3 - 3*x^4 - 2*x^5 + 4*x^6 + 9*x^7 + 3*x^8 - 15*x^9 +...
A201837 gives the real part of coefficients in 1/(1 - i*x - i*x^2) and begins: 1, 0, -1, -2, 0, 4, 5, -2, -13, -12, 12, 40, 25, -52, -117, -38, 196, 324,... in which the pairwise sums generate this sequence.

Paul D. Hanna, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (0,-1,-2,-1).

Cf. A201837 (real), A105309 (norm).

LinearRecurrence[{0,-1,-2,-1},{0,1,1,-1},50] (* Harvey P. Dale, Apr 23 2024 *)

{a(n)=imag(polcoeff(1/(1-I*x-I*x^2+x*O(x^n)),n))}

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

G.f.: x*(1+x)/(1 + x^2*(1+x)^2).
a(n) = A201837(n-1) + A201837(n-2), where A201837 gives the real part of the coefficients in 1/(1 - i*x - i*x^2).
a(n) = Im((((i + sqrt(4*i-1))^(n+1) - (i - sqrt(4*i-1))^(n+1)))/(2^(n+1)*sqrt(4*i-1))), where i=sqrt(-1). - Daniel Suteu, Apr 20 2018

A119749 Number of compositions of n into odd blocks with one element in each block distinguished.

Original entry on oeis.org

1, 1, 4, 7, 15, 32, 65, 137, 284, 591, 1231, 2560, 5329, 11089, 23076, 48023, 99935, 207968, 432785, 900633, 1874236, 3900319, 8116639, 16890880, 35150241, 73148321, 152223044, 316779047, 659223215, 1371856032, 2854858465

Offset: 1

Louis Shapiro, Jul 30 2006

The sequence is the INVERT transform of the aerated odd integers. - Gary W. Adamson, Feb 02 2014

Number of compositions of n into odd parts where there is 1 sort of part 1, 3 sorts of part 3, 5 sorts of part 5, ... , 2*k-1 sorts of part 2*k-1. - Joerg Arndt, Aug 04 2014

			a(3) = 4 since Abc, aBc, abC come from one block of size 3 and A/B/C comes from having three blocks. The capital letters are the distinguished elements.

R. X. F. Chen and L. W. Shapiro, On Sequences G(n) satisfying G(n)=(d+2)*G(n-1)-G(n-2), J. Int. Seq. 10 (2007) #07.8.1, Theorem 16.
Y-h. Guo, Some n-Color Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq. (6).
Y.-h. Guo, n-Color Odd Self-Inverse Compositions, J. Int. Seq. 17 (2014) # 14.10.5, eq. (2).
B. Hopkins, Spotted tilings and n-color compositions, INTEGERS 12B (2012/2013), #A6.
Index entries for linear recurrences with constant coefficients, signature (1,2,1,-1).

Cf. A105309, A052530, A000045, A030267. Row sums of A292835.

Rest@ CoefficientList[ Series[x(1 + x^2)/(x^4 - x^3 - 2x^2 - x + 1), {x, 0, 50}], x] (* Robert G. Wilson v *)

G.f.: (x+x^3)/(x^4 - x^3 -2x^2 -x +1).
a(n) = A092886(n)+A092886(n-2). - R. J. Mathar, Mar 08 2018
Sum_{k=0..n} a(k) = (3*a(n) + 2*a(n-1) - a(n-3))/2 - 1. - Xilin Wang and Greg Dresden, Aug 27 2020

A218135 Norm of coefficients in the expansion of 1 / (1 - x - 2Ix^2), where I^2=-1.

Original entry on oeis.org

1, 1, 5, 17, 45, 185, 533, 1921, 6205, 20745, 69541, 229585, 769613, 2552537, 8515125, 28340513, 94357853, 314301865, 1046284741, 3484682865, 11602442605, 38636214649, 128653931093, 428398492865, 1426535718525, 4750159951433, 15817576773605, 52670623373329

Offset: 0

Paul D. Hanna, Oct 21 2012

nonn

The radius of convergence of g.f. equals (1 + sqrt(65) - sqrt(2)*sqrt(1+sqrt(65)))/16 = 0.30031050...

			G.f.: A(x) = 1 + 4*x + 17*x^2 + 80*x^3 + 369*x^4 + 1700*x^5 + 7841*x^6 +...
The terms equal the norm of the complex coefficients in the expansion:
1/(1-x-2*I*x^2) = 1 + x + (1 + 2*I)*x^2 + (1 + 4*I)*x^3 + (-3 + 6*I)*x^4 + (-11 + 8*I)*x^5 + (-23 + 2*I)*x^6 + (-39 - 20*I)*x^7 + (-43 - 66*I)*x^8 +...
so that
a(1) = 1, a(2) = 1 + 2^2, a(3) = 1 + 4^2, a(4) = 3^2 + 6^2, a(5) = 11^2 + 8^2, ...

Cf. A105309, A218134.

{a(n)=norm(polcoeff(1/(1-x-2*I*x^2+x*O(x^n)), n))}
for(n=0,30,print1(a(n),", "))

G.f.: (1-4*x^2) / (1 - x - 8*x^2 - 4*x^3 + 16*x^4).

A218137 Sum of absolute values of real and imaginary parts of the coefficients in the expansion of 1 / (1 - x - I*x^2), where I^2=-1.

Original entry on oeis.org

1, 1, 2, 3, 3, 6, 9, 11, 16, 27, 37, 40, 77, 117, 144, 207, 351, 482, 523, 999, 1522, 1879, 2681, 4560, 6279, 6839, 12960, 19799, 24517, 34722, 59239, 81793, 89424, 168123, 257547, 319880, 449667, 769547, 1065430, 1169193, 2180881, 3350074, 4173363, 5823117, 9996480

Offset: 0

Paul D. Hanna, Oct 21 2012

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 3*x^4 + 6*x^5 + 9*x^6 + 11*x^7 + 16*x^8 +...
The terms equal the sum of absolute values of real and imaginary parts of the coefficients in the expansion:
1/(1-x-I*x^2) = 1 + x + (1 + I)*x^2 + (1 + 2*I)*x^3 + 3*I*x^4 + (-2 + 4*I)*x^5 + (-5 + 4*I)*x^6 + (-9 + 2*I)*x^7 + (-13 - 3*I)*x^8 + (-15 - 12*I)*x^9 + (-12 - 25*I)*x^10 - 40*I*x^11 + (25 - 52*I)*x^12 + (65 - 52*I)*x^13 + (117 - 27*I)*x^14 + (169 + 38*I)*x^15 + (196 + 155*I)*x^16 + (158 + 324*I)*x^17 + (3 + 520*I)*x^18 + (-321 + 678*I)*x^19 + (-841 + 681*I)*x^20 +...
so that
a(1) = 1, a(2) = 1 + 1, a(3) = 1 + 2, a(4) = 3, a(5) = 2 + 4, ...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A105309, A218138.

{a(n)=local(Cn=polcoeff(1/(1-x-I*x^2+x*O(x^n)),n));abs(real(Cn)) + abs(imag(Cn))}
for(n=0,40,print1(a(n),", "))

A171064 G.f.: -x(x-1)(1+x)/(1-x-7*x^2-x^3+x^4).

Original entry on oeis.org

0, 1, 1, 7, 15, 64, 175, 631, 1905, 6433, 20224, 66529, 212625, 692119, 2226799, 7217728, 23284815, 75343591, 243328225, 786800449, 2542156800, 8217744577, 26556314401, 85835882791, 277405671375, 896595420736, 2897714688751

Offset: 0

R. J. Mathar, at the request of R. K. Guy, Sep 03 2010

The member k=7 of a family of sequences starting 0,1,1,k with recurrence a(n) = a(n-1)+k*a(n-2)+a(n-3)-a(n-4).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hugh Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277
Index entries for linear recurrences with constant coefficients, signature (1,7,1,-1).

Cf. A116201 (k=1), A105309 (k=2), A152090 (k=3), A007598 (k=4), A005178 (k=5), A003757 (k=6).

I:=[0, 1, 1, 7]; [n le 4 select I[n] else Self(n-1) + 7*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[-x*(x - 1)*(1 + x)/(1 - x - 7*x^2 - x^3 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)
LinearRecurrence[{1,7,1,-1},{0,1,1,7},30] (* Harvey P. Dale, Nov 15 2020 *)

a(n) = +a(n-1) +7*a(n-2) +a(n-3) -a(n-4).

The roots (r1..r4) of the characteristic polynomials for this "family" of sequences have the following form (not simplified) for k= 1,2,3,4,5,6.... r1=(sqrt(4*k+10+2*sqrt(4*k+9))+sqrt(4*k-6+2*sqrt(4*k+9)))/4. r2=(sqrt(4*k+10+2*sqrt(4*k+9))-sqrt(4*k-6+2*sqrt(4*k+9)))/4. r3=(-sqrt(4*k+10-2*sqrt(4*k+9))-sqrt(4*k-6-2*sqrt(4*k+9)))/4. r4=(-sqrt(4*k+10-2*sqrt(4*k+9))+sqrt(4*k-6-2*sqrt(4*k+9)))/4. For k=1,2,3, r3 and r4 are complex . Closed-form (not simplified) is as follows for all k (note:for k1-k3 set r3 and r4 =0 and round a(n) to nearest integer): a(n)=sqrt(4*k+9)/(4*k+9)*(((r1)^n+(r2)^n)-((r3)^n+(r4)^n)). [Tim Monahan, Sep 17 2011]

A171065 G.f. -x(x-1)(1+x)/(1-x-8*x^2-x^3+x^4).

Original entry on oeis.org

0, 1, 1, 8, 17, 81, 224, 881, 2737, 9928, 32481, 113761, 380800, 1313441, 4441121, 15215688, 51677297, 176530481, 600723424, 2049428881, 6980069457, 23799693448, 81088954561, 276417142721, 941948403200, 3210574806081

Offset: 0

R. J. Mathar, at the request of R. K. Guy, Sep 03 2010

The member k=8 of a family of sequences starting 0,1,1,k with recurrence a(n) = a(n-1)+k*a(n-2)+a(n-3)-a(n-4).

This is the case P1 = 1, P2 = -10, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 31 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hugh Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory vol. 7 (5) (2011) 1255-1277
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume
Index entries for linear recurrences with constant coefficients, signature (1,8,1,-1).

Cf. A116201 (k=1), A105309 (k=2), A152090 (k=3), A007598 (k=4), A005178 (k=5), A003757 (k=6). A100047.

I:=[0, 1, 1, 8]; [n le 4 select I[n] else Self(n-1) + 8*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[-x*(x - 1)*(1 + x)/(1 - x - 8*x^2 - x^3 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)
LinearRecurrence[{1,8,1,-1},{0,1,1,8},30] (* Harvey P. Dale, Dec 27 2017 *)

a(n)= +a(n-1) +8*a(n-2) +a(n-3) -a(n-4).
From Peter Bala, Mar 31 2014: (Start)
a(n) = ( T(n,alpha) - T(n,beta) )/(alpha - beta), where alpha = (1 + sqrt(41))/4 and beta = (1 - sqrt(41))/4 and T(n,x) denotes the Chebyshev polynomial of the first kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, 5/2; 1, 1/2].
a(n) = U(n-1,i*(1 + sqrt(2))/2)*U(n-1,i*(1 + sqrt(2))/2), where U(n,x) denotes the Chebyshev polynomial of the second kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials and 4th-order linear divisibility sequences. (End)

A171066 G.f. -x(x-1)(1+x)/(1-x-9*x^2-x^3+x^4).

Original entry on oeis.org

0, 1, 1, 9, 19, 100, 279, 1189, 3781, 14661, 49600, 184141, 641421, 2333629, 8240959, 29700900, 105561739, 378777169, 1350292761, 4835148121, 17260998400, 61748847081, 220582688041, 788748162049, 2818480203099, 10076047502500

Offset: 0

R. J. Mathar, at the request of R. K. Guy, Sep 03 2010

The member k=9 of a family of sequences starting 0,1,1,k with recurrence a(n) = a(n-1)+k*a(n-2)+a(n-3)-a(n-4).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hugh Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory vol. 7 (5) (2011) 1255-1277
Index entries for linear recurrences with constant coefficients, signature (1,9,1,-1).

Cf. A116201 (k=1), A105309 (k=2), A152090 (k=3), A007598 (k=4), A005178 (k=5), A003757 (k=6).

I:=[0, 1, 1, 9]; [n le 4 select I[n] else Self(n-1) + 9*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[-x*(x - 1)*(1 + x)/(1 - x - 9*x^2 - x^3 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)

a(n)= +a(n-1) +9*a(n-2) +a(n-3) -a(n-4)

A171067 G.f. -x(x-1)(1+x)/((x^2+3x+1)(x^2-4*x+1)).

Original entry on oeis.org

0, 1, 1, 10, 21, 121, 340, 1561, 5061, 20890, 72721, 285121, 1028160, 3931201, 14425201, 54480250, 201635301, 756931801, 2813339860, 10529812921, 39218508021, 146573045290, 546474598561, 2040893746561, 7612994269440

Offset: 0

R. J. Mathar, at the request of R. K. Guy, Sep 03 2010

The member k=10 of a family of sequences starting 0,1,1,k with recurrence a(n) = a(n-1)+k*a(n-2)+a(n-3)-a(n-4).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Hugh Williams, R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory vol. 7 (5) (2011) 1255-1277
Index entries for linear recurrences with constant coefficients, signature (1,10,1,-1).

Cf. A116201 (k=1), A105309 (k=2), A152090 (k=3), A007598 (k=4), A005178 (k=5), A003757 (k=6).

I:=[0, 1, 1, 10]; [n le 4 select I[n] else Self(n-1) + 10*Self(n-2) + Self(n-3) - Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 19 2012

CoefficientList[Series[-x*(x - 1)*(1 + x)/((x^2 + 3*x + 1)*(x^2 - 4*x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 19 2012 *)
LinearRecurrence[{1,10,1,-1},{0,1,1,10},30] (* Harvey P. Dale, Dec 24 2017 *)

a(n)= +a(n-1) +10*a(n-2) +a(n-3) -a(n-4).

a(n)= -(-1)^n*A005248(n)/7 + 2*A001075(n)/7.

cp's OEIS Frontend

A218134 Norm of coefficients in the expansion of 1/(1 - 2*x - i*x^2), where i is the imaginary unit.

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A201837 G.f.: real part of 1/(1 - i*x - i*x^2) where i=sqrt(-1).

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A201838 G.f.: imaginary part of 1/(1 - i*x - i*x^2) where i=sqrt(-1).

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A119749 Number of compositions of n into odd blocks with one element in each block distinguished.

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A218135 Norm of coefficients in the expansion of 1 / (1 - x - 2*I*x^2), where I^2=-1.

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A218137 Sum of absolute values of real and imaginary parts of the coefficients in the expansion of 1 / (1 - x - I*x^2), where I^2=-1.

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A171064 G.f.: -x*(x-1)*(1+x)/(1-x-7*x^2-x^3+x^4).

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A218134 Norm of coefficients in the expansion of 1/(1 - 2x - ix^2), where i is the imaginary unit.

A201837 G.f.: real part of 1/(1 - ix - ix^2) where i=sqrt(-1).

A201838 G.f.: imaginary part of 1/(1 - ix - ix^2) where i=sqrt(-1).

A218135 Norm of coefficients in the expansion of 1 / (1 - x - 2Ix^2), where I^2=-1.

A171064 G.f.: -x(x-1)(1+x)/(1-x-7*x^2-x^3+x^4).

A171065 G.f. -x(x-1)(1+x)/(1-x-8*x^2-x^3+x^4).

A171066 G.f. -x(x-1)(1+x)/(1-x-9*x^2-x^3+x^4).

A171067 G.f. -x(x-1)(1+x)/((x^2+3x+1)(x^2-4*x+1)).