A106852 -id:A106852 - cp's OEIS frontend

A146078 Expansion of 1/(1-x(1-9x)).

1, 1, -8, -17, 55, 208, -287, -2159, 424, 19855, 16039, -162656, -307007, 1156897, 3919960, -6492113, -41771753, 16657264, 392603041, 242687665, -3290739704, -5474928689, 24141728647, 73416086848, -143859470975, -804604252607

Offset: 0

Philippe Deléham, Oct 27 2008

Row sums of Riordan array (1, x(1-9x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-9).

Cf. A010892, A107920, A106852, A106853, A106854, A145934, A145976, A145978.

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 9*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2018

LinearRecurrence[{1, -9}, {1, 1}, 100] (* G. C. Greubel, Jan 30 2016 *)

x='x+O('x^30); Vec(1/(1-x+9*x^2)) \\ G. C. Greubel, Jan 19 2018

[lucas_number1(n,1,9) for n in range(1, 27)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1) - 9*a(n-2), a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A109466(n,k)*9^(n-k).
From G. C. Greubel, Jan 31 2016: (Start)
G.f.: 1/(1-x+9*x^2).
E.g.f.: exp(x/2)*(cos(sqrt(35)*x/2) + (1/sqrt(35))*sin(sqrt(35)*x/2)). (End)
a(n) = Product_{k=1..n} (1 + 6*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019
a(n) = 3^n * U(n, 1/6), where U(n, x) is the Chebyshev polynomial of the second kind. - Federico Provvedi, Mar 28 2022

A214733 a(n) = -a(n-1) - 3*a(n-2) with n>1, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, -1, -2, 5, 1, -16, 13, 35, -74, -31, 253, -160, -599, 1079, 718, -3955, 1801, 10064, -15467, -14725, 61126, -16951, -166427, 217280, 282001, -933841, 87838, 2713685, -2977199, -5163856, 14095453, 1396115, -43682474, 39494129, 91553293, -210035680

Offset: 0

Roman Witula, Jul 27 2012

The sequence a(n) is conjugate with A110523 by the following alternative relations: either ((-1 + i*sqrt(11))/2)^n = A110523(n) + a(n)*(-1 + i*sqrt(11))/2, or ((-1 - i*sqrt(11))/2)^n = A110523(n) + a(n)*(-1 - i*sqrt(11))/2 (see also comments to A110523, where these relations and many other facts on a(n) is presented).
Apart from signs, the Lucas U(P=1,Q=3)-sequence. - R. J. Mathar, Oct 24 2012
This is the Lucas U(-1, 3) sequence. (V_n(-1, 3))^2 + 11*(U_n(-1, 3))^2 = 4*Q^n = 4*3^n. For the special case where |U_n(-1, 3)| = 1, then, by the Lucas sequence identity U_2*n = U_n*V_n, we have (U_2*n(-1, 3))^2 + 11 = 4*3^n, true for n = 1, 2, 5, U_n = 1, -1, 1 and U_2*n = -1, 5, -31. E.g., (-31)^2 + 11 = 972 = 4*3^5. - Raphie Frank, Dec 09 2015

Roman Witula, On Some Applications of Formulae for Unimodular Complex Numbers, Jacek Skalmierski's Press, Gliwice 2011.

Seiichi Manyama, Table of n, a(n) for n = 0..4189
Ronald Orozco López, Deformed Differential Calculus on Generalized Fibonacci Polynomials, arXiv:2211.04450 [math.CO], 2022.
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (-1,-3).

Cf. A106852, A110523.

[n le 2 select n-1 else -Self(n-1)-3*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Dec 10 2015

LinearRecurrence[{-1, -3}, {0, 1}, 40] (* T. D. Noe, Jul 30 2012 *)

concat(0,Vec(1/(1+x+3*x^2)+O(x^99))) \\ Charles R Greathouse IV, Oct 01 2012

[(-1)^(n-1)*round(3^((n-1)/2)*chebyshev_U(n-1, 1/(2*sqrt(3)))) for n in range(41)] # G. C. Greubel, Dec 28 2023

a(n) = - a(n-1) - 3*a(n-2).
a(n) = (-1)^n*(i*sqrt(11)/11)*(((1 + i*sqrt(11))/2)^n - ((1 - i*sqrt(11))/2)^n).
G.f.: x/(1 + x + 3*x^2).
G.f.: Q(0) -1, where Q(k) = 1 - 3*x^2 - (k+2)*x + x*(k+1 + 3*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 07 2013
From G. C. Greubel, Dec 28 2023: (Start)
a(n) = (-1)^(n-1)*3^((n-1)/2)*ChebyshevU(n-1, 1/(2*sqrt(3))).
a(n) = (-1)^n * A106852(n-1).
E.g.f.: (2/sqrt(11))*exp(-x/2)*sin(sqrt(11)*x/2). (End)

A146083 Expansion of 1/(1 - x(1 - 11x)).

Original entry on oeis.org

1, 1, -10, -21, 89, 320, -659, -4179, 3070, 49039, 15269, -524160, -692119, 5073641, 12686950, -43123101, -182679551, 291674560, 2301149621, -907270539, -26219916370, -16239940441, 272179139629, 450818484480, -2543152051439

Offset: 0

Philippe Deléham, Oct 27 2008

Row sums of Riordan array (1,x(1-11x)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-11).

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

[n le 2 select 1 else Self(n-1)-11*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jan 31 2016

LinearRecurrence[{1,-11},{1,1},30] (* Harvey P. Dale, Jan 07 2016 *)

Vec(1/(1-x*(1-11*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

[lucas_number1(n,1,11) for n in range(1, 26)] # Zerinvary Lajos, Apr 22 2009

a(n) = a(n-1)-11*a(n-2) ; a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A109466(n,k)*11^(n-k).
From G. C. Greubel, Jan 31 2016: (Start)
G.f.: 1/(1-x+11*x^2).
E.g.f.: exp(x/2)*(cos(sqrt(43)*x/2) + (1/sqrt(43))*sin(sqrt(43)*x/2)).
(End)

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).

A106855 Expansion of 1/(1-x^2(1-3x)).

Original entry on oeis.org

1, 0, 1, -3, 1, -6, 10, -9, 28, -39, 55, -123, 172, -288, 541, -804, 1405, -2427, 3817, -6642, 11098, -18093, 31024, -51387, 85303, -144459, 239464, -400368, 672841, -1118760, 1873945, -3137283, 5230225, -8759118, 14642074, -24449793, 40919428, -68376015, 114268807, -191134299, 319396852

Offset: 0

Paul Barry, May 08 2005

Diagonal sums of Riordan array (1,x(1-3x)).

Index entries for linear recurrences with constant coefficients, signature (0,1,-3).

Cf. A052947, A106852.

G.f.: 1/(1-x^2+3x^3); a(n)=a(n-2)-3a(n-3); a(n)=sum{k=0..floor(n/2), (-1)^n*binomial(k, n-2k)*3^(n-2k)}.

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

A190959 a(n) = 3a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 3, 4, -3, -29, -72, -71, 147, 796, 1653, 979, -5328, -20879, -35997, -3596, 169197, 525571, 730728, -435671, -4960653, -12703604, -13307547, 23595379, 137323872, 293994721, 195364803, -883879196, -3628461603, -6465988829, -1255658472, 28562968729

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

This is the Lucas U(P=3, Q=5) sequence. - R. J. Mathar, Oct 24 2012

a(n+2)/a(n+1) equals the continued fraction 3 - 5/(3 - 5/(3 - 5/(3 - ... - 5/3))) with n 5's. - Greg Dresden, Oct 06 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Wikipedia, Lucas sequence
Index entries for linear recurrences with constant coefficients, signature (3,-5).

Cf. A190958 (index to generalized Fibonacci sequences), A190970 (binomial transf.), A106852 (inv. bin. transf., shifted).

I:=[0,1]; [n le 2 select I[n] else 3*Self(n-1) - 5*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 25 2018

Mathematica
```
LinearRecurrence[{3,-5}, {0,1}, 50]
```

x='x+O('x^30); concat([0], Vec(x/(1-3x+5*x^2))) \\ G. C. Greubel, Jan 25 2018

G.f.: x/(1 - 3*x + 5*x^2). - Philippe Deléham, Oct 11 2011

E.g.f.: 2*exp(3*x/2)*sin(sqrt(11)*x/2)/sqrt(11). - Stefano Spezia, Oct 06 2019

A115672 Coefficients of L-series for elliptic curve "35a3": y^2 + y = x^3 + x^2 - x.

Original entry on oeis.org

1, 0, 1, -2, -1, 0, 1, 0, -2, 0, -3, -2, 5, 0, -1, 4, 3, 0, 2, 2, 1, 0, -6, 0, 1, 0, -5, -2, 3, 0, -4, 0, -3, 0, -1, 4, 2, 0, 5, 0, -12, 0, -10, 6, 2, 0, 9, 4, 1, 0, 3, -10, 12, 0, 3, 0, 2, 0, 0, 2, 8, 0, -2, -8, -5, 0, -4, -6, -6, 0, 0, 0, 2, 0, 1, -4, -3, 0, -1, -4, 1, 0, 12, -2, -3, 0, 3, 0, -12, 0, 5, 12, -4, 0, -2, 0, -1, 0, 6, -2, 6

Offset: 1

Michael Somos, Jan 29 2006

			q + q^3 - 2*q^4 - q^5 + q^7 - 2*q^9 - 3*q^11 - 2*q^12 + 5*q^13 - q^15 + ...

Robin Visser, Table of n, a(n) for n = 1..10000
LMFDB, Elliptic curve with LMFDB label 35.a2 (Cremona label 35a3)
W. Stein, Modular Forms Database.

Cf. A106852(n) = a(3^n).

{a(n)=if( n<1, 0, ellak( ellinit([ 0, 1, 1, -1, 0]), n))} /* Michael Somos, Mar 03 2011 */

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^5 + A) * eta(x^7 + A))^2 + x^2 * (eta(x + A) * eta(x^35 + A))^2, n))}

def a(n):
    return EllipticCurve("35a3").an(n)  # Robin Visser, Sep 30 2023

a(n) is multiplicative with a(5^e) = (-1)^e, a(7^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) otherwise.

Expansion of (eta(q^5) * eta(q^7))^2 + (eta(q) * eta(q^35))^2 in powers of q. Expansion of a newform level 35 weight 2 and trivial character.

A201947 Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 2, 0, 3, 5, 1, -1, 5, 10, 4, -2, -1, 8, 20, 12, -4, -4, 0, 13, 38, 31, -4, -13, -2, 1, 21, 71, 73, 3, -33, -11, 3, 1, 34, 130, 162, 34, -74, -42, 6, 6, 0, 55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Offset: 0

Philippe Deléham, Dec 06 2011

Row-reversed variant of A123585. Row sums: 2^n.

			Triangle begins:
1
1, 1
2, 2, 0
3, 5, 1, -1
5, 10, 4, -2, -1
8, 20, 12, -4, -4, 0
13, 38, 31, -4, -13, -2, 1
21, 71, 73, 3, -33, -11, 3, 1
34, 130, 162, 34, -74, -42, 6, 6, 0
55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Cf. Columns: A000045, A001629, A129707.

Diagonals: A010892, A099254, Antidiagonal sums: A158943.

G.f.: 1/(1-(1+y)*x+(y+1)*(y-1)*x^2).
T(n,0) = A000045(n+1).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-2) with T(0,0)= 1 and T(n,k)= 0 if nSum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A090591(n), (-1)^n*A106852(n), A000007(n), A000045(n+1), A000079(n), A057083(n), A190966(n+1) for n = -3, -2, -1, 0, 1, 2, 3 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A010892(n), A000079(n), A030195(n+1), A180222(n+2) for x = 0, 1, 2, 3 respectively.

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cp's OEIS Frontend

A146078 Expansion of 1/(1-x*(1-9*x)).

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A214733 a(n) = -a(n-1) - 3*a(n-2) with n>1, a(0)=0, a(1)=1.

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A146083 Expansion of 1/(1 - x*(1 - 11*x)).

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

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A106855 Expansion of 1/(1-x^2(1-3x)).

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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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A190959 a(n) = 3*a(n-1) - 5*a(n-2), with a(0)=0, a(1)=1.

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A146078 Expansion of 1/(1-x(1-9x)).

A146083 Expansion of 1/(1 - x(1 - 11x)).

A190959 a(n) = 3a(n-1) - 5a(n-2), with a(0)=0, a(1)=1.