A099450 -id:A099450 - cp's OEIS frontend

A099451 A Chebyshev transform of A099450 associated to the knot 7_7.

1, 5, 17, 45, 96, 155, 119, -365, -2217, -7360, -18791, -38435, -57639, -28875, 200992, 1015075, 3179711, 7796715, 15240559, 20915840, 3218033, -103746315, -458355231, -1362884995, -3211177504, -5977952405, -7345234233, 2382397955, 51340513351, 204512766400, 579756435849

Offset: 0

Paul Barry, Oct 16 2004

The denominator is a parameterization of the Alexander polynomial for the knot 7_7. The g.f. is the image of the g.f. of A099450 under the Chebyshev transform A(x)->(1/(1+x^2))A(x/(1+x^2)).

Dror Bar-Natan, The Rolfsen Knot Table
Index entries for linear recurrences with constant coefficients, signature (5,-9,5,-1).

Cf. A099450, A099452.

G.f.: (1+x^2)/(1-5x+9x^2-5x^3+x^4).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k * Sum_{j=0..n-2k} C(n-2k-j, j)*(-7)^j*5^(n-2k-2j).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*A099450(n-2k).
a(n) = Sum_{k=0..n} binomial((n+k)/2, k)*(-1)^((n-k)/2)*(1+(-1)^(n+k))*A099450(k)/2.
a(n) = Sum_{k even, 0<=k<=n} A099452(n-k). [corrected by Kevin Ryde, Jul 24 2022]

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

A367300 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2x - x^2.

Original entry on oeis.org

1, 3, 2, 10, 10, 3, 33, 46, 22, 4, 109, 194, 131, 40, 5, 360, 780, 678, 296, 65, 6, 1189, 3036, 3228, 1828, 581, 98, 7, 3927, 11546, 14514, 10100, 4194, 1036, 140, 8, 12970, 43150, 62601, 51664, 26479, 8604, 1722, 192, 9, 42837, 159082, 261598, 249720, 152245, 61318, 16248, 2712, 255, 10

Offset: 1

Clark Kimberling, Dec 23 2023

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
     3      2
    10     10      3
    33     46     22      4
   109    194    131     40     5
   360    780    678    296    65     6
  1189   3036   3228   1828   581    98    7
  3927  11546  14514  10100  4194  1036  140  8
Row 4 represents the polynomial p(4,x) = 33 + 46*x + 22*x^2 + 4*x^3, so (T(4,k)) = (33,46,22,4), k=0..3.

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

Cf. A006190 (column 1); A000027 (p(n,n-1)); A107839 (row sums, p(n,1)); A001045 (alternating row sums, p(n,-1)); A030240 (p(n,2)); A039834 (signed Fibonacci numbers, p(n,-2)); A016130 (p(n,3)); A225883 (p(n,-3)); A099450 (p(n,-4)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299.

p[1, x_] := 1; p[2, x_] := 3 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - 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) = 3 + 2*x, u = p(2,x), and v = 1 - 2*x - x^2.

p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(13 + 4*x)), b = (1/2) (2*x + 3 + 1/k), c = (1/2) (2*x + 3 - 1/k).

A099452 An Alexander sequence for the knot 7_7.

Original entry on oeis.org

1, 5, 16, 40, 79, 110, 23, -520, -2336, -6995, -16574, -31075, -38848, 9560, 258631, 1043950, 2978719, 6781640, 12060848, 13119125, -12022526, -124662155, -461573264, -1259138680, -2752822273, -4615067410, -4134056729, 8360350360, 58685747584, 202130368445, 528415922498

Offset: 0

Paul Barry, Oct 16 2004

The denominator is a parameterization of the Alexander polynomial for the knot 7_7. 1/(1-5*x+9*x^2-5*x^3+x^4) is the image of the g.f. of A099450 under the modified Chebyshev transform A(x)->(1/(1+x^2)^2)A(x/(1+x^2)).

Dror Bar-Natan, The Rolfsen Knot Table
Index entries for linear recurrences with constant coefficients, signature (5,-9,5,-1).

LinearRecurrence[{5,-9,5,-1},{1,5,16,40,79},40] (* Harvey P. Dale, Apr 18 2019 *)

G.f.: (1-x)*(1+x)*(1+x^2)/(1-5*x+9*x^2-5*x^3+x^4). - corrected by R. J. Mathar, Nov 24 2012

a(n)=A099451(n)-A099451(n-2).

cp's OEIS Frontend

A099451 A Chebyshev transform of A099450 associated to the knot 7_7.

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

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A367300 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 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 - 2*x - x^2.

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A099452 An Alexander sequence for the knot 7_7.

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Mathematica

Formula

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

A367300 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 2x, p(n,x) = up(n-1,x) + vp(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2x - x^2.