A099459 -id:A099459 - cp's OEIS frontend

A099460 A Chebyshev transform of A099459 associated to the knot 9_48.

1, 7, 39, 203, 1040, 5313, 27133, 138565, 707643, 3613904, 18456077, 94254531, 481354555, 2458260679, 12554250288, 64114111901, 327428500337, 1672165762785, 8539691368807, 43611901581472, 222724437852585

Offset: 0

Paul Barry, Oct 16 2004

The denominator is a parameterization of the Alexander polynomial for the knot 9_48. The g.f. is the image of the g.f. of A099459 under the Chebyshev transform A(x) -> (1/(1+x^2))*A(x/(1+x^2)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Dror Bar-Natan, 9 48, The Knot Atlas.
Index entries for linear recurrences with constant coefficients, signature (7,-11,7,-1).

Cf. A099459, A099461.

I:=[1,7,39,203]; [n le 4 select I[n] else 7*Self(n-1) - 11*Self(n-2) +7*Self(n-3) -Self(n-4): n in [1..31]]; // G. C. Greubel, Nov 18 2021

LinearRecurrence[{7,-11,7,-1}, {1,7,39,203}, 30] (* G. C. Greubel, Nov 18 2021 *)

def A099460_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2)/(1-7*x+11*x^2-7*x^3+x^4) ).list()
A099460_list(30) # G. C. Greubel, Nov 18 2021

G.f.: (1+x^2)/(1 -7*x +11*x^2 -7*x^3 +x^4).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*( Sum_{j=0..n-2*k} C(n-2*k-j, j)(-9)^j*7^(n-2*k-2*j) ).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)(-1)^k*A099459(n-2*k).
a(n) = (1/2)*Sum_{k=0..n} (-1)^((n-k)/2)*(1 + (-1)^(n+k))*binomial((n+k)/2, k) *A099459(k).
a(n) = Sum_{k=0..n} A099461(n-k)*binomial(1, k/2)*((1+(-1)^k)/2).

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

Original entry on oeis.org

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

A099461 An Alexander sequence for the knot 9_48.

Original entry on oeis.org

1, 7, 38, 196, 1001, 5110, 26093, 133252, 680510, 3475339, 17748434, 90640627, 462898478, 2364006148, 12072895733, 61655851222, 314874250049, 1608051650884, 8212262868470, 41939735818687, 214184746483778, 1093833919809295, 5586171115205846, 28528378178106436, 145693417671662033, 744051127629095062, 3799842775146922277, 19405662567631938052, 99104031922539424718

Offset: 0

Paul Barry, Oct 16 2004

The denominator 1 -7*x +11*x^2 -7*x^3 +x^4 is a parameterization of the Alexander polynomial for the knot 9_48. 1/(1 -7*x +11*x^2 -7*x^3 +x^4) is the image of the g.f. of A099459 under the modified Chebyshev transform A(x) -> (1/(1+x^2)^2)*A(x/(1+x^2)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Dror Bar-Natan, 9 48, The Knot Atlas.
Index entries for linear recurrences with constant coefficients, signature (7,-11,7,-1).

Cf. A099459, A099460.

I:=[7,38,196,1001]; [1] cat [n le 4 select I[n] else 7*Self(n-1) - 11*Self(n-2) +7*Self(n-3) -Self(n-4): n in [1..41]]; // G. C. Greubel, Nov 18 2021

LinearRecurrence[{7,-11,7,-1},{1,7,38,196,1001},40] (* Harvey P. Dale, Jun 18 2021 *)

def A099461_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)*(1+x)*(1+x^2)/(1-7*x+11*x^2-7*x^3+x^4) ).list()
A099461_list(40) # G. C. Greubel, Nov 18 2021

a(n) = A099460(n) - A099460(n-2).

G.f.: (1-x)*(1+x)*(1+x^2)/(1-7*x+11*x^2-7*x^3+x^4). - Corrected by R. J. Mathar, Nov 23 2012

A102902 a(n) = 9a(n-1) - 16a(n-2), with a(0) = 1, a(1) = 9.

Original entry on oeis.org

1, 9, 65, 441, 2929, 19305, 126881, 833049, 5467345, 35877321, 235418369, 1544728185, 10135859761, 66507086889, 436390025825, 2863396842201, 18788331166609, 123280631024265, 808912380552641, 5307721328585529

Offset: 0

Paul Barry, Jan 17 2005

Indranil Ghosh, Table of n, a(n) for n = 0..1221
R. Flórez, R. A. Higuita, and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5, Journal of Integer Sequences, Vol. 17 (2014).
Index entries for linear recurrences with constant coefficients, signature (9,-16).

Cf. A002450, A004187, A016153, A099459.

[4^n*Evaluate(ChebyshevSecond(n+1), 9/8): n in [0..30]]; // G. C. Greubel, Dec 09 2022

LinearRecurrence[{9,-16},{1,9},20] (* Harvey P. Dale, Jul 28 2016 *)

[lucas_number1(n,9,16) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

G.f.: 1/(1-9*x+16*x^2).
a(n) = Sum_{k=0..n} binomial(2*n-k+1, k)*4^k.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-16)^k*9^(n-2*k).
a(n) = 4^n * ChebyshevU(n, 9/8). - G. C. Greubel, Dec 09 2022
From Peter Bala, Jul 23 2025: (Start)
a(n) := ((9 + sqrt(17))^(n+1) - (9 - sqrt(17))^(n+1))/(2^(n+1)*sqrt(17)).
The following products telescope:
Product_{k >= 1} 1 + 4^k/a(k) = (1 + sqrt(17))/2.
Product_{k >= 1} 1 - 4^k/a(k) = (1 + sqrt(17))/18.
Product_{k >= 1} 1 + (-4)^k/a(k) = (17 + sqrt(17))/34.
Product_{k >= 1} 1 - (-4)^k/a(k) = (17 + sqrt(17))/18. (End)

cp's OEIS Frontend

A099460 A Chebyshev transform of A099459 associated to the knot 9_48.

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Formula

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

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A099461 An Alexander sequence for the knot 9_48.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A102902 a(n) = 9*a(n-1) - 16*a(n-2), with a(0) = 1, a(1) = 9.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

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

A102902 a(n) = 9a(n-1) - 16a(n-2), with a(0) = 1, a(1) = 9.