A107920 -id:A107920 - cp's OEIS frontend

A105579 a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 3, a(2) = 4.

1, 3, 4, 1, -4, -3, 8, 17, 4, -27, -32, 25, 92, 45, -136, -223, 52, 501, 400, -599, -1396, -195, 2600, 2993, -2204, -8187, -3776, 12601, 20156, -5043, -45352, -35263, 55444, 125973, 15088, -236855, -267028, 206685, 740744, 327377, -1154108, -1808859, 499360, 4117081, 3118364, -5115795, -11352520

Offset: 0

Creighton Dement, Apr 14 2005

Floretion Algebra Multiplication Program, FAMP Code: famseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e]

Cf. A002249, A014551, A078020, A105577, A105578, A105580.

Cf. Equals (1/2) [A107920(n+4) - 2*A107920(n-1) + 3 ].

Table[(3 - ((1-I*Sqrt[7])^n + (1+I*Sqrt[7])^n)/2^n)/2 // Simplify, {n, 1, 50}] (* Jean-François Alcover, Jun 04 2017 *)

a(n+1) - a(n) = A002249(n).

a(n) = 2*a(n-1)-3*a(n-2)+2*a(n-3). G.f.: (1+x+x^2)/((1-x)*(1-x+2*x^2)). [Colin Barker, Mar 27 2012]

Corrected by T. D. Noe, Nov 07 2006

A146084 Expansion of 1/(1-x(1-12x)).

Original entry on oeis.org

1, 1, -11, -23, 109, 385, -923, -5543, 5533, 72049, 5653, -858935, -926771, 9380449, 20501701, -92063687, -338084099, 766680145, 4823689333, -4376472407, -62260744403, -9743075519, 737385857317, 854302763545, -7994327524259

Offset: 0

Philippe Deléham, Oct 27 2008

Row sums of Riordan array (1,x(1-12x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1, -12).

Cf. A010892, A107920, A106852, A106853, A106854, A145934, A145976, A145978, A146078, A146080, A146083

LinearRecurrence[{1, -12}, {1, 1}, 100] (* G. C. Greubel, Jan 30 2016 *)

Vec(1/(1-x*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[lucas_number1(n,1,12) for n in range(1, 26)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1)-12*a(n-2), a(0)=1, a(1)=1.

a(n) = Sum_{k, 0<=k<=n} A109466(n,k)*12^(n-k).

A128100 Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 5, 1, 8, 10, 3, 13, 20, 9, 1, 21, 38, 22, 4, 34, 71, 51, 14, 1, 55, 130, 111, 40, 5, 89, 235, 233, 105, 20, 1, 144, 420, 474, 256, 65, 6, 233, 744, 942, 594, 190, 27, 1, 377, 1308, 1836, 1324, 511, 98, 7, 610, 2285, 3522, 2860, 1295, 315, 35, 1, 987, 3970

Offset: 0

Emeric Deutsch, Feb 18 2007

Row sums are the Jacobsthal numbers (A001045). Column 0 yields the Fibonacci numbers (A000045); the other columns yield convolved Fibonacci numbers (A001629, A001628, A001872, A001873, etc.). Sum_{k=0..floor(n/2)} k*T(n,k) = A073371(n-2).
Triangle T(n,k), with zeros omitted, given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 24 2012
Riordan array (1/(1-x-x^2), x^2/(1-x-x^2)), with zeros omitted. - Philippe Deléham, Feb 06 2012
Diagonal sums are A000073(n+2) (tribonacci numbers). - Philippe Deléham, Feb 16 2014
Number of induced subgraphs of the Fibonacci cube Gamma(n-1) that are isomorphic to the hypercube Q_k. Example: row n=4 is 5, 5, 1; indeed, the Fibonacci cube Gamma(3) is a square with an additional pendant edge attached to one of its vertices; it has 5 vertices (i.e., Q_0's), 5 edges (i.e., Q_1's) and 1 square (i.e., Q_2). - Emeric Deutsch, Aug 12 2014
Row n gives the coefficients of the polynomial p(n,x) defined as the numerator of the rational function given by f(n,x) = 1 + (x + 1)/f(n-1,x), where f(x,0) = 1. Conjecture: for n > 2, p(n,x) is irreducible if and only if n is a (prime - 2). - Clark Kimberling, Oct 22 2014

			Triangle starts:
   1;
   1;
   2,  1;
   3,  2;
   5,  5,  1;
   8, 10,  3;
  13, 20,  9,  1;
  21, 38, 22,  4;
From _Philippe Deléham_, Jan 24 2012: (Start)
Triangle (1, 1, -1, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, ...) begins:
   1;
   1,  0;
   2,  1,  0;
   3,  2,  0,  0;
   5,  5,  1,  0,  0;
   8, 10,  3,  0,  0,  0;
  13, 20,  9,  1,  0,  0,  0;
  21, 38, 22,  4,  0,  0,  0,  0; (End)
From _Clark Kimberling_, Oct 22 2014: (Start)
Here are the first 4 polynomials p(n,x) as in Comment and generated by Mathematica program:
  1
  2 +  x
  3 + 2x
  5 + 5x + x^2. (End)

C.-P. Chou and H. A. Witek, An Algorithm and FORTRAN Program for Automatic Computation of the Zhang-Zhang Polynomial of Benzenoids, MATCH: Commun. Math. Comput. Chem, 68 (2012) 3-30. See Eq. (9). - From _N. J. A. Sloane_, Dec 23 2012
S. Klavzar, M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, preprint.
S. Klavzar, M. Mollard, Cube polynomial of Fibonacci and Lucas cubes, Acta Appl. Math. 117, 2012, 93-105. - _Emeric Deutsch_, Aug 12 2014

Cf. A001045, A000045, A001629, A001628, A001872, A001873, A073371.

Cf. A109466, A119473.

G:=1/(1-z-(1+t)*z^2): Gser:=simplify(series(G,z=0,19)): for n from 0 to 16 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 16 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form

p[x_, n_] := 1 + (x + 1)/p[x, n - 1]; p[x_, 1] = 1;
Numerator[Table[Factor[p[x, n]], {n, 1, 20}]]  (* Clark Kimberling, Oct 22 2014 *)

G.f.: 1/(1-z-(1+t)z^2).
Sum_{k=0..n} T(n,k)*x^k = A053404(n), A015447(n), A015446(n), A015445(n), A015443(n), A015442(n), A015441(n), A015440(n), A006131(n), A006130(n), A001045(n+1), A000045(n+1), A000012(n), A010892(n), A107920(n+1), A106852(n), A106853(n), A106854(n), A145934(n), A145976(n), A145978(n), A146078(n), A146080(n), A146083(n), A146084(n) for x = 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, and -13, respectively. - Philippe Deléham, Jan 24 2012
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1). - Philippe Deléham, Jan 24 2012
G.f.: T(0)/2, where T(k) = 1 + 1/(1 - (2*k+1+ x*(1+y))*x/((2*k+2+ x*(1+y))*x + 1/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013
T(n,k) = Sum_{i=k..floor(n/2)} binomial(n-i,i)*binomial(i,k). See Corollary 3.3 in the Klavzar et al. link. - Emeric Deutsch, Aug 12 2014

A192395 a(n) = 2a(n-1) - 2a(n-2) + a(n-3) + 2*a(n-4) starting with a(0..3) = 0, 0, 0, 1.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 1, 2, 8, 17, 22, 22, 33, 78, 156, 233, 298, 442, 833, 1546, 2464, 3553, 5390, 9230, 16161, 26358, 40404, 62713, 103298, 174290, 285505, 451154, 712184, 1156145, 1910086, 3122374, 5005089, 7987806, 12907980, 21090185, 34362394, 55428010

Offset: 0

Paul Curtz, Jun 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,1,2).

Cf. A000045, A107920.

I:=[0,0,0,1]; [n le 4 select I[n] else 2*Self(n-1)-2*Self(n-2)+Self(n-3)+2*Self(n-4): n in [1..50]]; // Vincenzo Librandi, Nov 25 2011

a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <2|1|-2|2>>^n)[1,4]:
seq(a(n), n=0..45);  # Alois P. Heinz, Jul 11 2023

LinearRecurrence[{2,-2,1,2},{0,0,0,1},100] (* Vincenzo Librandi, Nov 25 2011 *)

@CachedFunction
def a(n): # a = A192395
    if (n<4): return n//3
    else: return 2*a(n-1) - 2*a(n-2) + a(n-3) + 2*a(n-4)
[a(n) for n in range(51)] # G. C. Greubel, Jul 10 2023

From R. J. Mathar, Jul 14 2011: (Start)
G.f.: x^3/((1-x+2*x^2)*(1-x-x^2)).
a(n) = (A000045(n) - A107920(n))/3. (End)

A340670 Number of complex base i-1 points which can be represented within n bits and negated within those n bits.

Original entry on oeis.org

1, 1, 1, 3, 5, 15, 29, 47, 101, 199, 413, 847, 1621, 3255, 6541, 13087, 26373, 52423, 104637, 209711, 419253, 839511, 1678317, 3353919, 6710629, 13421287, 26845213, 53693007, 107366933, 214742391, 429498701, 858994271, 1718023109, 3435955975, 6871883645

Offset: 0

Kevin Ryde, Jan 15 2021

Complex base i-1 of Khmelnik and Penney uses an integer m to represent a complex integer point z(m) = A318438(m) + A318439(m)*i.  A340669(m) is the negation of z in this representation.  a(n) is how many n-bit m are negatable within those n bits, i.e., how many m in the range 0 <= m < 2^n have also 0 <= A340669(m) < 2^n.
A geometric interpretation of a(n) is to draw a unit square around each point z(0) to z(2^n-1), rotate a copy by 180 degrees about the origin, and measure its area of intersection with the original.
The bit-flip rule in A340669 gives the recurrence formula below.  A low 0-bit of m has a(n-1) negatables above it, or low 11 is one arbitrary bit then negatables above so 2*a(n-3), or low 01 is three arbitrary so 8*a(n-5).  This can be thought of the number of compositions of n (partitions with order) into parts 1,3,5 with 2 types of part 3 and 8 types of part 5.

			For n=3, the a(3)=3 points of n bits are m = 0,3,7 < 2^n, which negate to A340669(0,3,7) = 0,7,3 < 2^n.  These m are located at z = 0,i,-i,
               negate        intersection
  z(0..7)    (rotate 180)   a(3) = 3 points
                * *
   * *            * *             *
     o *        * o               o
   * *            * *             *
     * *

Kevin Ryde, Table of n, a(n) for n = 0..700
Kevin Ryde, Iterations of the Dragon Curve, see index MinusNegA.
Index entries for linear recurrences with constant coefficients, signature (1,0,2,0,8).

Cf. A340669, A107920, A078069.

{ my(table=[4,-2,-2,6, -4,2,2,-6], p=Mod('x,2-'x+'x^2));
a(n) = (6<

a(n) = a(n-1) + 2*a(n-3) + 8*a(n-5).
a(n) = (2/5)*2^n + (h/15)*2^floor(n/2) + (2/3)*Im((1/2 + i*sqrt(7)/2)^(n+1)) where h = 4,-2,-2,6, -4,2,2,-6 according as n == 0 to 7 (mod 8) respectively.
a(n) = (2/5)*2^n + (1/3)*A107920(n+1) + (1/15)*A078069(n+2).
G.f.: 1/(1 - x - 2*x^3 - 8*x^5).
G.f.: (2/5)/(1-2*x) + (1/3)/(1-x+2*x^2) + (2/15)*(2+3*x)/(1+2*x+2*x^2).

A332332 Coefficients of L-series for elliptic curve "33a1": y^2 + xy = x^3 + x^2 - 11x.

Original entry on oeis.org

1, 1, -1, -1, -2, -1, 4, -3, 1, -2, 1, 1, -2, 4, 2, -1, -2, 1, 0, 2, -4, 1, 8, 3, -1, -2, -1, -4, -6, 2, -8, 5, -1, -2, -8, -1, 6, 0, 2, 6, -2, -4, 0, -1, -2, 8, 8, 1, 9, -1, 2, 2, 6, -1, -2, -12, 0, -6, -4, -2, 6, -8, 4, 7, 4, -1, -4, 2, -8, -8, 0, -3, -14, 6

Offset: 1

Michael Somos, Feb 23 2020

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

Robin Visser, Table of n, a(n) for n = 1..10000
LMFDB, Elliptic Curve with LMFDB label 33.a2 (Cremona label 33a1)

Cf. A107920, A168175.

A := Basis( ModularForms( Gamma0(33), 2), 75); A[2] + A[3] - A[4] - A[5] - A[6];

{a(n) = if( n<1, 0, ellak( ellinit( [1, 1, 0, -11, 0], 1), n))};

{a(n) = my(A, t1, t3); if( n<1, 0, n--; A = x * O(x^n); t1 = eta(x + A) * eta(x^11 + A); t3 = x * eta(x^3 + A) * eta(x^33 + A); polcoeff( t1^2 + 3*t1*t3 + 3*t3^2, n))};

def a(n):
    return EllipticCurve("33a1").an(n)  # Robin Visser, Sep 30 2023

Expansion of eta(q)^2*eta(q^11)^2 + 3*eta(q)*eta(q^3)*eta(q^11)*eta(q^33) + 3*eta(q^3)^2*eta(q^33)^2 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (33 t)) = 33 (t/i)^2 f(t) where q = exp(2 Pi i t).
a(n) is multiplicative with a(3^n) = (-1)^n, a(11^n) = 1. a(2^n) = A107920(n+1). a(7^n) = A168175(n).

A077876 Expansion of (1-x)^(-1)/(1-x+2*x^2).

Original entry on oeis.org

1, 2, 1, -2, -3, 2, 9, 6, -11, -22, 1, 46, 45, -46, -135, -42, 229, 314, -143, -770, -483, 1058, 2025, -90, -4139, -3958, 4321, 12238, 3597, -20878, -28071, 13686, 69829, 42458, -97199, -182114, 12285, 376514, 351945, -401082, -1104971, -302806, 1907137, 2512750, -1301523, -6327022

Offset: 0

N. J. A. Sloane, Nov 17 2002

Row sums of Riordan array (1/(1-x),x(1-2x)). - Paul Barry, Jul 18 2005

Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (2,-3,2).

Cf. A001045, A107920.

Join[{a=1,b=2},Table[c=b-2*a+1;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)

Vec(1/((1-x)*(1-x+2*x^2)) + O(x^75)) \\ Michel Marcus, Feb 08 2015

a(n) = Sum_{k=0..n} U(k, 1/(2*sqrt(2)))*sqrt(2)^k. - Paul Barry, Nov 20 2003 [There is a missing parenthesis, but it is not clear where it should be inserted. - N. J. A. Sloane, Feb 08 2015] [corrected by Jason Yuen, Aug 22 2024]
a(n) = Sum_{k=2..n-2} A107920(k). - R. J. Mathar, Oct 13 2017
a(n) = Sum_{k=0..floor((n+1)/2)} binomial(n+1-k,k+1)*(-2)^k,  n>=0. - Taras Goy, Apr 15 2020

A105576 a(n) = 2a(n-1) - 3a(n-2) + 2*a(n-3) with a(0) = 3, a(1) = 4, a(2) = 0.

Original entry on oeis.org

3, 4, 0, -6, -4, 10, 20, 2, -36, -38, 36, 114, 44, -182, -268, 98, 636, 442, -828, -1710, -52, 3370, 3476, -3262, -10212, -3686, 16740, 24114, -9364, -57590, -38860, 76322, 154044, 1402, -306684, -309486, 303884, 922858, 315092, -1530622, -2160804, 900442, 5222052, 3421170

Offset: 0

Creighton Dement, Apr 14 2005

Floretion Algebra Multiplication Program, FAMP Code: 1vesseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e]

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

Cf. A105577, A105578, A105579, A105580.

Equals 2*A107920(n) + A107920(n-1) + 1.

LinearRecurrence[{2,-3,2},{3,4,0},50] (* Harvey P. Dale, Jul 05 2022 *)

2*a(n) = A105225(n) + A105577(n) + 4*((-1)^n)*A001607(n+1)

G.f.: (3-2x+x^2)/((1-x)(1-x+2x^2)). a(n)=1+A107920(n)+2*A107920(n+1). [From R. J. Mathar, Feb 04 2009]

A191897 Coefficients of the Z(n,x) polynomials; Z(0,x) = 1, Z(1,x) = x and Z(n,x) = xZ(n-1,x) - 2Z(n-2,x), n >= 2.

Original entry on oeis.org

1, 1, 0, 1, 0, -2, 1, 0, -4, 0, 1, 0, -6, 0, 4, 1, 0, -8, 0, 12, 0, 1, 0, -10, 0, 24, 0, -8, 1, 0, -12, 0, 40, 0, -32, 0, 1, 0, -14, 0, 60, 0, -80, 0, 16, 1, 0, -16, 0, 84, 0, -160, 0, 80, 0, 1, 0, -18, 0, 112, 0, -280, 0, 240, 0, -32

Offset: 0

Paul Curtz, Jun 19 2011

The coefficients of the Z(n,x) polynomials by decreasing exponents, see the formulas, define this triangle.

			The first few rows of the coefficients of the Z(n,x) are
  1;
  1,    0;
  1,    0,   -2;
  1,    0,   -4,    0;
  1,    0,   -6,    0,    4;
  1,    0,   -8,    0,   12,    0;
  1,    0,  -10,    0,   24,    0,   -8;
  1,    0,  -12,    0,   40,    0,  -32,    0;
  1,    0,  -14,    0,   60,    0,  -80,    0,   16;
  1,    0,  -16,    0,   84,    0, -160,    0,   80,    0;

Row sums: A107920(n+1). Main diagonal: A077966(n).
Z(n,x=1) = A107920(n+1), Z(n,x=2)  = A009545(n+1),
Z(n,x=3) = A000225(n+1), Z(n,x=4)  = A007070(n),
Z(n,x=5) = A107839(n),   Z(n,x=6)  = A154244(n),
Z(n,x=7) = A186446(n),   Z(n,x=8)  = A190975(n+1),
Z(n,x=9) = A190979(n+1), Z(n,x=10) = A190869(n+1).
Row sum without sign: A113405(n+1).
Cf. A128099, A013609.

nmax:=10: Z(0, x):=1 : Z(1, x):=x: for n from 2 to nmax do Z(n, x) := x*Z(n-1, x) - 2*Z(n-2, x) od: for n from 0 to nmax do for k from 0 to n do T(n, k) := coeff(Z(n, x), x, n-k) od: od: seq(seq(T(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jun 27 2011, revised Nov 29 2012

a[n_, k_] := If[OddQ[k], 0, 2^(k/2)*Coefficient[ ChebyshevU[n, x/2], x, n-k]]; Flatten[ Table[ a[n, k], {n, 0, 10}, {k, 0, n}]] (* Jean-François Alcover, Aug 02 2012, from 2nd formula *)

Z(0,x) = 1, Z(1,x) = x and Z(n,x) = x*Z(n-1,x) - 2*Z(n-2,x), n >= 2.

a(n,k) = A077957(k) * A053119(n,k). - Paul Curtz, Sep 30 2011

Edited and information added by Johannes W. Meijer, Jun 27 2011

A228439 Numbers k dividing u(k), where the Lucas sequence is defined u(i) = u(i-1) - 2*u(i-2) with initial conditions u(0)=0, u(1)=1.

Original entry on oeis.org

1, 7, 49, 343, 2401, 4753, 16807, 33271, 76783, 117649, 232897, 461041, 537481, 823543, 1630279, 3227287, 3762367, 5764801, 7447951, 11411953, 11527201, 19358969, 22591009, 26336569, 40353607, 44720977, 52135657, 79883671, 80690407

Offset: 1

Thomas M. Bridge, Nov 02 2013

Since the absolute value of the discriminant of the characteristic polynomial is prime (=7), the sequence contains every nonnegative integer power of 7. Other terms are formed on multiplication of 7^k by sporadic primes.

			For k = 0, 1 , ..., 10, there is u(k) = 0,1,1,-1,-3,-1,5,7,-3,-17,-11. Clearly only k = 1 and k = 7 satisfy k divides u(k).

Chris Smyth, The Terms in Lucas Sequences Divisible by their Indices, Journal of Integer Sequences, Vol. 13 (2010), Article 10.2.4.
Wikipedia, Lucas sequence.

Cf. A107920 (Lucas Sequence u(n)=u(n-1)-2u(n-2)).

nn = 10000; s = LinearRecurrence[{1, -2}, {1, 1}, nn]; t = {}; Do[If[Mod[s[[n]], n] == 0, AppendTo[t, n]], {n, nn}]; t (* T. D. Noe, Nov 08 2013 *)

a(19)-a(29) from Amiram Eldar, May 28 2024

cp's OEIS Frontend

A105579 a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 3, a(2) = 4.

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A146084 Expansion of 1/(1-x(1-12x)).

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A128100 Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).

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A192395 a(n) = 2*a(n-1) - 2*a(n-2) + a(n-3) + 2*a(n-4) starting with a(0..3) = 0, 0, 0, 1.

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A340670 Number of complex base i-1 points which can be represented within n bits and negated within those n bits.

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A332332 Coefficients of L-series for elliptic curve "33a1": y^2 + x*y = x^3 + x^2 - 11*x.

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A077876 Expansion of (1-x)^(-1)/(1-x+2*x^2).

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A192395 a(n) = 2a(n-1) - 2a(n-2) + a(n-3) + 2*a(n-4) starting with a(0..3) = 0, 0, 0, 1.

A332332 Coefficients of L-series for elliptic curve "33a1": y^2 + xy = x^3 + x^2 - 11x.

A105576 a(n) = 2a(n-1) - 3a(n-2) + 2*a(n-3) with a(0) = 3, a(1) = 4, a(2) = 0.

A191897 Coefficients of the Z(n,x) polynomials; Z(0,x) = 1, Z(1,x) = x and Z(n,x) = xZ(n-1,x) - 2Z(n-2,x), n >= 2.