A190970 -id:A190970 - cp's OEIS frontend

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

0, 1, 2, -6, -32, -4, 312, 664, -1792, -10224, -2528, 97184, 219648, -532544, -3261568, -1197696, 30220288, 72417536, -157367808, -1038910976, -504143872, 9380822016, 23803082752, -46202054656, -330434936832, -198849327104, 2906650714112, 7801794699264

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

For the difference equation a(n) = c*a(n-1) - d*a(n-2), with a(0) = 0, a(1) = 1, the solution is a(n) = d^((n-1)/2) * ChebyshevU(n-1, c/(2*sqrt(d))) and has the alternate form a(n) = ( ((c + sqrt(c^2 - 4*d))/2)^n - ((c - sqrt(c^2 - 4*d))/2)^n )/sqrt(c^2 - 4*d). In the case c^2 = 4*d then the solution is a(n) = n*d^((n-1)/2). The generating function is x/(1 - c*x + d^2) and the exponential generating function takes the form (2/sqrt(c^2 - 4*d))*exp(c*x/2)*sinh(sqrt(c^2 - 4*d)*x/2) for c^2 > 4*d, (2/sqrt(4*d - c^2))*exp(c*x/2)*sin(sqrt(4*d - c^2)*x/2) for 4*d > c^2, and x*exp(sqrt(d)*x) if c^2 = 4*d. - G. C. Greubel, Jun 10 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..190
Index entries for linear recurrences with constant coefficients, signature (2,-10).

Sequences of the form a(n) = c*a(n-1) - d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A128834,A107920,A106852,A106853,A106854,A145934,A145976,A145978,A146078,A146080
.A000027,A009545,A088137,A088138,A045873,A088139,A089181,A090591,A127357,A190958
.A001906,A000225,A057083,A049072,A190959,A190960,A190961,A190962,A190963,A190964
.A001353,A007070,A003462,A001787,A099456,A190965,A168175,A190966,A190967,A190968
.A004254,A107839,A116415,A002450,A030191,A001047,A099450,A190969,A190970,A190971
.A001109,A154244,A138395,A084326,A003463,A030192,A081179,A006516,A027471,A106392
.A004187,A186446,A190972,A190973,A190974,A003464,A030240,A152268,A099459,A016127
.A001090,A190975,A190976,A099156,A190977,A190978,A023000,A057084,A154245,A154235
.A018913,A190979,A190980,A190981,A190982,A190983,A190984,A023001,A057085,A178869
A004189,A190869,A190985,A190986,A190987,A190988,A190989,A190990,A002452,A057086

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1)-10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 17 2011

Mathematica
```
LinearRecurrence[{2,-10}, {0,1}, 50]
```

a(n)=([0,1; -10,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

[lucas_number1(n,2,10) for n in (0..50)] # G. C. Greubel, Jun 10 2022

G.f.: x / ( 1 - 2*x + 10*x^2 ). - R. J. Mathar, Jun 01 2011
E.g.f.: (1/3)*exp(x)*sin(3*x). - Franck Maminirina Ramaharo, Nov 13 2018
a(n) = 10^((n-1)/2) * ChebyshevU(n-1, 1/sqrt(10)). - G. C. Greubel, Jun 10 2022
a(n) = (1/3)*10^(n/2)*sin(n*arctan(3)) = Sum_{k=0..floor(n/2)} (-1)^k*3^(2*k)*binomial(n,2*k+1). - Gerry Martens, Oct 15 2022

A368153 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3x - x^2.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 4, -2, 4, 5, 5, 4, -10, 5, 8, 10, -3, 4, -25, 6, 13, 16, 1, -29, 14, -49, 7, 21, 28, -8, -24, -78, 56, -84, 8, 34, 47, -12, -88, -26, -162, 168, -132, 9, 55, 80, -31, -140, -200, 100, -330, 408, -195, 10, 89, 135, -58, -301, -230, -296

Offset: 1

Clark Kimberling, Jan 20 2024

Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

			First eight rows:
   1
   1   2
   2   1   3
   3   4  -2    4
   5   5   4  -10    5
   8  10  -3    4  -25    6
  13  16   1  -29   14  -49    7
  21  28  -8  -24  -78   56  -84   8
Row 4 represents the polynomial p(4,x) = 3 + 4*x - 2*x^2 + 4*x^3, so (T(4,k)) = (3,4,-2,4), k=0..3.

Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), 1-28, Paper No. A14.

Cf. A000045 (column 1); A000027 (p(n,n-1)); A057083 (row sums), (p(n,1)); A182228 (alternating row sums), (p(n,-1)); A190970, (p(n,2)); A030195, (p(n,-2)); A052918, (p(n,-3)); A190972, (p(n,-4)); A057085, (p(n,-5)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152.

p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 3x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 2*x, u = p(2,x), and v = 1 - 3*x - x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 - 8*x), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

A103644 Expansion of g.f. (3x+1)/((1-3x)(1+5x+9x^2)).

Original entry on oeis.org

1, 1, 4, 25, 1, 256, 169, 1225, 5476, 961, 64009, 25600, 358801, 1164241, 515524, 15642025, 3243601, 101284096, 239228089, 216825625, 3736387876, 287336401, 27697946329, 47210598400

Offset: 0

Creighton Dement, Feb 11 2005

A floretion-generated sequence of squares.

This sequence is also related to several other sequences of squares.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Creighton Dement, Online Floretion Multiplier. [broken link]
Index entries for linear recurrences with constant coefficients, signature (-2,6,27).

Cf. A103645.

A103644 := proc(n)
    6*3^n+5*(-1)^n*A190970(n+1)+18*(-1)^(n+1)*A190970(n) ;
    %/11 ;
end proc:
seq(A103644(n),n=0..20) ; # R. J. Mathar, Mar 23 2023

CoefficientList[Series[(3x+1)/(1+2x-6x^2-27x^3),{x,0,30}],x] (* or *) LinearRecurrence[{-2,6,27},{1,1,4},30] (* Harvey P. Dale, Dec 13 2017 *)

a(n+3) = -2a(n+2) + 6a(n+1) + 27a(n), a(0) = 1, a(1) = 1, a(2) = 4.
a(n) = (1/11)*(2*3^n-(-5/2-(I*sqrt(11))/2)^n-(-5/2+(I*sqrt(11))/2)^n). [Creighton Dement, May 24 2009]
11*a(n) = 6*3^n + 5*b(n) + 18*b(n-1) where b(n) = (-1)^n*A190970(n+1). - R. J. Mathar, Mar 23 2023

Corrected by T. D. Noe, Nov 07 2006

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

A087169 Expansion of (1 + 3x)/(1 + 5x + 9*x^2).

Original entry on oeis.org

1, -2, 1, 13, -74, 253, -599, 718, 1801, -15467, 61126, -166427, 282001, 87838, -2977199, 14095453, -43682474, 91553293, -64624199, -500858642, 3085911001, -10921827227, 26835937126, -35883240587, -62107231199, 633485321278, -2608461525599, 7340939736493, -13228544952074

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 22 2003

For positive n, a(n) equals 3^n times the permanent of the (2n) X (2n) tridiagonal matrix with 1/sqrt(3)'s along the main diagonal, and i's along the superdiagonal and the subdiagonal (where i is the imaginary unit). - John M. Campbell, Jul 08 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-5,-9).

Cf. A190970.

[n le 2 select (-1)^(n-1)*n else -5*Self(n-1) - 9*Self(n-2): n in [1..51]]; // G. C. Greubel, Jun 09 2022

CoefficientList[Series[(1+3x)/(1+5x+9x^2), {x, 0, 30}], x]
LinearRecurrence[{-5,-9},{1,-2},50] (* Harvey P. Dale, Mar 24 2021 *)

def A087169(n): return (-3)^n*(chebyshev_U(n, 5/6) - chebyshev_U(n-1, 5/6))
[A087169(n) for n in (0..50)] # G. C. Greubel, Jun 09 2022

G.f.: (1 + 3*x)/(1 + 5*x + 9*x^2).
a(n) = -5*a(n-1) - 9*a(n-2), a(0)=1, a(1)=-2.
a(n) = Sum_{k=0..n} binomial(n+k,2*k)*(-3)^(n-k).
a(n) = (-3)^n*( ChebyshevU(n, 5/6) - ChebyshevU(n-1, 5/6) ). - G. C. Greubel, Jun 09 2022

cp's OEIS Frontend

A190958 a(n) = 2*a(n-1) - 10*a(n-2), with a(0) = 0, a(1) = 1.

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A368153 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3*x - x^2.

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A103644 Expansion of g.f. (3x+1)/((1-3*x)*(1+5*x+9*x^2)).

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A087169 Expansion of (1 + 3*x)/(1 + 5*x + 9*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

A368153 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3x - x^2.

A103644 Expansion of g.f. (3x+1)/((1-3x)(1+5x+9x^2)).

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

A087169 Expansion of (1 + 3x)/(1 + 5x + 9*x^2).