A214733 -id:A214733 - cp's OEIS frontend

A106852 Expansion of 1/(1-x(1-3x)).

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, -64624199, -694731239, -500858642

Offset: 0

Paul Barry, May 08 2005

Row sums of Riordan array (1, x*(1-3*x)). In general, Sum_{k=0..n} (-1)^(n-k)*binomial(k,n-k)*r^(n-k) yields the row sums of the Riordan array (1, x(1-r*x)).
Row sums of Riordan array (1/(1+3*x^2), x/(1+3*x^2)). - Paul Barry, Sep 10 2005
See A214733 for a differently signed version of this sequence. - Peter Bala, Nov 21 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-3).

Cf. A214733.

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 3*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018

CoefficientList[Series[1/(1 - x (1 - 3 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 07 2013 *)
LinearRecurrence[{1,-3},{1,1},40] (* Harvey P. Dale, Apr 02 2016 *)

a(n)=([0,1; -3,1]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Nov 21 2016

x='x+O('x^30); Vec(1/(1-x+3*x^2)) \\ G. C. Greubel, Jan 14 2018

[lucas_number1(n,1,+3) for n in range(1, 40)] # Zerinvary Lajos, Apr 22 2009

From Paul Barry, Sep 10 2005: (Start)
G.f.: 1/(1-x+3*x^2).
a(n) = 2*sqrt(33)*3^(n/2)*cos((n+1)*arctan(sqrt(11)/11)-pi*n/2)/11.
a(n) = 3^(n/2)(cos(-n*arccot(sqrt(11)/11))-sqrt(11)*sin(-n*arccot(sqrt(11)/11))/11).
a(n) = ((1+sqrt(-11))^(n+1)-(1-sqrt(-11))^(n+1))/(2^(n+1)sqrt(-11)).
a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k, n-k)*3^(n-k) = Sum_{k=0..n} A109466(n,k)*3^(n-k).
a(n) = Sum_{k=0..n} C((n+k)/2, k)*(-3)^((n-k)/2)*(1+(-1)^(n-k))/2.
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)(-3)^k. (End)
a(n) = a(n-1) - 3*a(n-2), a(0)=1, a(1)=1. - Philippe Deléham, Oct 21 2008
G.f.: Q(0)/x -1/x, 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
G.f.: Sum_{n >= 0} x^n * Product_{k = 1..n} (k - 3*x)/(1 + k*x). - Peter Bala, Jul 06 2025

A110523 Expansion of (1 + x)/(1 + x + 3*x^2).

Original entry on oeis.org

1, 0, -3, 3, 6, -15, -3, 48, -39, -105, 222, 93, -759, 480, 1797, -3237, -2154, 11865, -5403, -30192, 46401, 44175, -183378, 50853, 499281, -651840, -846003, 2801523, -263514, -8141055, 8931597, 15491568, -42286359, -4188345, 131047422, -118482387, -274659879, 630107040, 193872597

Offset: 0

Paul Barry, Jul 24 2005

Row sums of number triangle A110522.

The sequence a(n) is conjugate with A214733 since the following alternative relations: either ((-1 + i*sqrt(11))/2)^n = a(n) + A214733(n)*(-1 + i*sqrt(11))/2 or ((-1 - i*sqrt(11))/2)^n = a(n) + A214733(n)*(-1 - i*sqrt(11))/2. We have a(n+1) = -3*A214733(n), A214733(n+1) = a(n) - A214733(n). We note that sequences A110512 and A001607 are conjugated in a similar way. The above relations are connected with the Gauss sums, for example if e := exp(i*2Pi/11) then e + e^3 + e^4 + e^5 + e^9 = (-1 + i*sqrt(11))/2, and e^2 + e^6 + e^7 + e^8 + e^10 = (-1 - i*sqrt(11))/2, which is equivalent to the system of sums: Sum_{k=1..5} cos(2Pi*k/11) = -1/2 and Sum_{k=1..5} sin(2Pi*k/11) = sqrt(11)/2, and which is equivalent to the system of products: P_{k=1..5} cos(2Pi*k/11) = -1/32 and P_{k=1..5} sin(2Pi*k/11) = sqrt(11)/32 - for details see Witula's book. Lastly we note that ((-1 + i*sqrt(11))/2)^n + ((-1 - i*sqrt(11))/2)^n = 2*a(n) - A214733(n). - Roman Witula, Jul 27 2012

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae, Vol. 38, No. 2 (2024), pp. 284-313. See p. 298.
Index entries for linear recurrences with constant coefficients, signature (-1,-3).

Cf. A001607, A106852, A110512, A110522, A214733.

[n le 2 select 2-n else -(Self(n-1) +3*Self(n-2)): n in [1..50]]; // G. C. Greubel, Dec 28 2023

CoefficientList[Series[(1+x)/(1+x+3*x^2), {x,0,50}], x] (* G. C. Greubel, Aug 30 2017 *)
LinearRecurrence[{-1,-3},{1,0},40] (* Harvey P. Dale, Jul 02 2022 *)

my(x='x+O('x^50)); Vec((1+x)/(1+x+3*x^2)) \\ G. C. Greubel, Aug 30 2017

@CachedFunction # a = A110523
def a(n): return 1-n if n<2 else -a(n-1) -3*a(n-2)
[a(n) for n in range(41)] # G. C. Greubel, Dec 28 2023

a(n) = Sum_{k=0..n} Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-3)^(j-k)*C(k, j-k).
From Roman Witula, Jul 27 2012: (Start)
a(n+2) + a(n+1) + 3*a(n) = 0.
a(n+1) = (-1)^n*(3*i*sqrt(11)/11)*(((1 + i*sqrt(11))/2)^(n-1) - ((1 - i*sqrt(11))/2)^(n-1)). (End)
From G. C. Greubel, Dec 28 2023: (Start)
a(n) = (-1)^n*3^((n-1)/2)*( sqrt(3)*ChebyshevU(n, 1/(2*sqrt(3))) - ChebyshevU(n-1, 1/(2*sqrt(3))) ).
a(n) = A106852(n) - A106852(n-1).
a(n) = (-1)^n*( A214733(n+1) + A214733(n) ). (End)
E.g.f.: exp(-x/2)*(sqrt(11)*cos(sqrt(11)*x/2) + sin(sqrt(11)*x/2))/sqrt(11). - Stefano Spezia, Jul 27 2025

A226075 Expansion of (eta(q) * eta(q^11))^2 + 2 * (eta(q^2) * eta(q^22))^2 in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, May 25 2013

			G.f. = q - q^3 - 2*q^4 + q^5 - 2*q^7 + 4*q^8 - 2*q^9 + q^11 + 2*q^12 + 4*q^13 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500

Cf. A006571, A090132, A214733.

Basis( CuspForms( Gamma0(22), 2), 79)[1];

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^11])^2 + 2 q^2 ( QPochhammer[ q^2] QPochhammer[ q^22])^2, {q, 0, n}]; (* Michael Somos, Apr 25 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^11 + A))^2 + 2 * x * (eta(x^2 + A) * eta(x^22 + A))^2, n))};

Sage
```
CuspForms( Gamma0(22), 2, prec=79).0;
```

a(n) is multiplicative with a(11^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) if p != 11.
G.f. is a period 1 Fourier series which satisfies f(-1 / (22 t)) = 22 (t/i)^2 f(t) where q = exp(2 Pi i t).
a(4*n + 2) = 0. a(4*n) = -2 * A006571(n). a(2^n) = A090132(n). a(3^n) = A214733(n+1).

A228440 Numbers n dividing u(n), where the Lucas sequence is defined u(i) = u(i-1) - 3*u(i-2) with initial conditions u(0)=0, u(1)=1.

Original entry on oeis.org

1, 11, 121, 253, 1331, 2783, 5819, 11891, 14641, 29161, 30613, 64009, 130801, 133837, 161051, 273493, 320771, 336743, 558877, 640343, 670703, 704099, 895873, 1438811, 1472207, 1771561, 3008423, 3078251, 3528481, 3544453, 3704173, 6147647, 6290339, 7027801

Offset: 1

Thomas M. Bridge, Nov 02 2013

nonn

Since the absolute value of the discriminant of the characteristic polynomial is prime (=11), the sequence contains every nonnegative integer power of 11 (A001020 is subsequence). Other terms are formed on multiplication of 11^k by sporadic primes.

			u(1)=1 and u(11)=253. Clearly n divides u(n) for these terms.

Lars Blomberg, Table of n, a(n) for n = 1..65
C. 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. A214733 (Lucas sequence u(n) ignoring sign).

Cf. A001020 (powers of 11).

nn = 10000; s = LinearRecurrence[{1, -3}, {1, 1}, nn]; t = {}; Do[
If[Mod[s[[n]], n] == 0, AppendTo[t, n]], {n, nn}]; t (* T. D. Noe, Nov 06 2013 *)

a(27)-a(34) from Lars Blomberg, Feb 15 2016

cp's OEIS Frontend

A106852 Expansion of 1/(1-x*(1-3*x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

A110523 Expansion of (1 + x)/(1 + x + 3*x^2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A226075 Expansion of (eta(q) * eta(q^11))^2 + 2 * (eta(q^2) * eta(q^22))^2 in powers of q.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A228440 Numbers n dividing u(n), where the Lucas sequence is defined u(i) = u(i-1) - 3*u(i-2) with initial conditions u(0)=0, u(1)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A106852 Expansion of 1/(1-x(1-3x)).