cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A098303 Member r=18 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 18, 289, 4608, 73441, 1170450, 18653761, 297289728, 4737981889, 75510420498, 1203428746081, 19179349516800, 305666163522721, 4871479266846738, 77638002106025089, 1237336554429554688
Offset: 0

Views

Author

Wolfdieter Lang, Oct 18 2004

Keywords

Links

Programs

  • Mathematica
    LinearRecurrence[{# - 1, -# + 1, 1}, {0, 1, #}, 17] &[18] (* Michael De Vlieger, Feb 23 2021 *)

Formula

a(n) = (T(n, 8)-1)/7 with Chebyshev's polynomials of the first kind evaluated at x=8: T(n, 8)=A001081(n)= ((8+3*sqrt(7))^n + (8-3*sqrt(7))^n)/2.
a(n) = 16*a(n-1) - a(n-2) + 2, n>=2, a(0)=0, a(1)=1.
a(n) = 17*a(n-1) - 17*a(n-2) + a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=18.
G.f.: x*(1+x)/((1-x)*(1-16*x+x^2)) = x*(1+x)/(1-17*x+17*x^2-x^3) (from the Stephan link, see A092184).

A098304 Member r=19 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 19, 324, 5491, 93025, 1575936, 26697889, 452288179, 7662201156, 129805131475, 2199025033921, 37253620445184, 631112522534209, 10691659262636371, 181127094942284100, 3068468954756193331
Offset: 0

Views

Author

Wolfdieter Lang, Oct 18 2004

Keywords

Links

Programs

  • Mathematica
    LinearRecurrence[{# - 1, -# + 1, 1}, {0, 1, #}, 17] &[19] (* Michael De Vlieger, Feb 23 2021 *)

Formula

a(n) = 2*(T(n, 17/2)-1)/15 with twice the Chebyshev polynomials of the first kind evaluated at x=17/2: 2*T(n, 17/2) = A078367(n) = ((17+sqrt(285))^n + (17-sqrt(285))^n)/2^n.
a(n) = 17*a(n-1) - a(n-2) + 2, n>=2, a(0)=0, a(1)=1.
a(n) = 18*a(n-1) - 18*a(n-2) + a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=19.
G.f.: x*(1+x)/((1-x)*(1-17*x+x^2)) = x*(1+x)/(1-18*x+18*x^2-x^3) (from the Stephan link, see A092184).

A098305 Unsigned member r=-5 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 5, 36, 245, 1681, 11520, 78961, 541205, 3709476, 25425125, 174266401, 1194439680, 8186811361, 56113239845, 384605867556, 2636127833045, 18068288963761, 123841894913280, 848824975429201, 5817932933091125, 39876705556208676, 273319005960369605, 1873356336166378561
Offset: 0

Views

Author

Wolfdieter Lang, Oct 18 2004

Keywords

Comments

((-1)^(n+1))*a(n) = S_{-5}(n), n>=0, defined in A092184.

Links

Crossrefs

Cf. A000032 (Lucas), A056854, A092184.

Formula

a(n) = 2*(T(n, 7/2)-(-1)^n)/9, with twice the Chebyshev polynomials of the first kind evaluated at x=7/2: 2*T(n, 7/2) = A056854(n) = ((7+sqrt(45))^n + (7-sqrt(45))^n)/2^n.
a(n) = 7*a(n-1)-a(n-2)+2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.
a(n) = 6*a(n-1) + 6*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=5.
G.f.: x*(1-x)/((1+x)*(1-7*x+x^2)) = x*(1-x)/(1-6*x-6*x^2+x^3) (from the Stephan link, see A092184).
a(n) = (Lucas(4*n) - 2*(-1)^n)/9. - Greg Dresden, Oct 10 2020

Extensions

More terms from Michel Marcus, Oct 11 2020

A098309 Unsigned member r = -10 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 10, 121, 1440, 17161, 204490, 2436721, 29036160, 345997201, 4122930250, 49129165801, 585427059360, 6975995546521, 83126519498890, 990542238440161, 11803380341783040, 140650021862956321, 1675996882013692810, 19971312562301357401, 237979753865602596000
Offset: 0

Views

Author

Wolfdieter Lang, Oct 18 2004

Keywords

Comments

((-1)^(n+1))*a(n) = S_{-10}(n), n>=0, defined in A092184.

Links

Programs

  • Mathematica
    LinearRecurrence[{11,11,-1},{0,1,10},30] (* Harvey P. Dale, Oct 28 2019 *)
  • PARI
    concat(0, Vec(x*(1-x)/(1-11*x-11*x^2+x^3) + O(x^30))) \\ Colin Barker, Jan 31 2017

Formula

a(n) = (T(n, 6)-(-1)^n)/7, with Chebyshev's polynomials of the first kind evaluated at x=6: T(n, 6)=A023038(n)=((6+sqrt(35))^n + (6-sqrt(35))^n)/2.
a(n) = 12*a(n-1)-a(n-2)+2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.
a(n) = 11*a(n-1) + 11*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=10.
G.f.: x*(1-x)/((1+x)*(1-12*x+x^2)) = x*(1-x)/(1-11*x-11*x^2+x^3) (from the Stephan link, see A092184).
a(n) = (-2*(-1)^n + (6-sqrt(35))^n + (6+sqrt(35))^n) / 14. - Colin Barker, Jan 31 2017

A099270 Unsigned member r=-12 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 12, 169, 2352, 32761, 456300, 6355441, 88519872, 1232922769, 17172398892, 239180661721, 3331356865200, 46399815451081, 646266059449932, 9001325016847969, 125372284176421632, 1746210653453054881
Offset: 0

Views

Author

Wolfdieter Lang, Oct 18 2004

Keywords

Comments

((-1)^(n+1))*a(n) = S_{-12}(n), n>=0, defined in A092184.

Links

Programs

  • Mathematica
    a[n_] := (ChebyshevT[n, 7] - (-1)^n)/8; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jun 21 2013, from 1st formula *)
CoefficientList[Series[x (1 - x) / ((1 + x) (1 - 14 x + x^2)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 21 2013 *)
  • PARI
    a(n)=real(((7+4*quadgen(12))^n-(-1)^n)/8) /* Michael Somos, Apr 30 2005 */
  • PARI
    a(n)=n=abs(2*n); round(2^(n-4)*prod(k=1,n,2-sin(2*Pi*k/n)))

Formula

a(n) = (T(n, 7)-(-1)^n)/8, with Chebyshev's polynomials of the first kind evaluated at x=7: T(n, 7)=A011943(n)=((7+4*sqrt(3))^n + (7-4*sqrt(3))^n)/2.
a(n) = 13*a(n-1) + 13*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=12.
G.f.: x*(1-x)/((1+x)*(1-14*x+x^2)) = x*(1-x)/(1-13*x-13*x^2+x^3) (from the Stephan link, see A092184).
a(n) = 14*a(n-1)-a(n-2)-2*(-1)^n, a(0)=0, a(1)=1. a(-n)=a(n).

A099272 Unsigned member r=-14 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 14, 225, 3584, 57121, 910350, 14508481, 231225344, 3685097025, 58730327054, 936000135841, 14917271846400, 237740349406561, 3788928318658574, 60385112749130625, 962372875667431424, 15337580897929772161, 244438921491208923150, 3895685162961412998241
Offset: 0

Views

Author

Wolfdieter Lang, Oct 18 2004

Keywords

Comments

((-1)^(n+1))*a(n) = S_{-14}(n), n>=0, defined in A092184.

Links

Crossrefs

Cf. A001081.

Programs

  • Maple
    f:= n -> (orthopoly[T](n,8)-(-1)^n)/9:
map(f, [$0..20]); # Robert Israel, Jun 04 2018
  • Mathematica
    CoefficientList[Series[x (1-x)/(1-15 x-15 x^2+x^3),{x,0,33}],x] (* Vincenzo Librandi, Jun 05 2018 *)

Formula

a(n) = (T(n, 8)-(-1)^n)/9, with Chebyshev's polynomials of the first kind evaluated at x=8: T(n, 8)=A001081(n)=((8+3*sqrt(7))^n + (8-3*sqrt(7))^n)/2.
a(n) = 16*a(n-1)-a(n-2)+2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.
a(n) = 15*a(n-1) + 15*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=14.
G.f.: x*(1-x)/((1+x)*(1-16*x+x^2)) = x*(1-x)/(1-15*x-15*x^2+x^3) (from the Stephan link, see A092184).

A098307 Unsigned member r=-7 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 7, 64, 567, 5041, 44800, 398161, 3538647, 31449664, 279508327, 2484125281, 22077619200, 196214447521, 1743852408487, 15498457228864, 137742262651287, 1224181906632721, 10879894897043200, 96694872166756081
Offset: 0

Views

Author

Wolfdieter Lang, Oct 18 2004

Keywords

Comments

((-1)^(n+1))*a(n) = S_{-7}(n), n>=0, defined in A092184.

Links

Programs

  • Mathematica
    LinearRecurrence[{8,8,-1},{0,1,7},20] (* Harvey P. Dale, Jan 01 2017 *)

Formula

a(n)= 2*(T(n, 9/2)-(-1)^n)/11, with twice Chebyshev's polynomials of the first kind evaluated at x=9/2: 2*T(n, 9/2)=A056918(n)=((9+sqrt(77))^n + (9-sqrt(77))^n)/2^n.
a(n)= 9*a(n-1)-a(n-2)+2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.
a(n)= 8*a(n-1) + 8*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=7.
G.f.: x*(1-x)/((1+x)*(1-9*x+x^2)) = x*(1-x)/(1-8*x-8*x^2+x^3) (from the Stephan link, see A092184).

A098310 Unsigned member r=-11 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 11, 144, 1859, 24025, 310464, 4012009, 51845651, 669981456, 8657913275, 111882891121, 1445819671296, 18683772835729, 241443227193179, 3120078180675600, 40319573121589619, 521034372399989449
Offset: 0

Views

Author

Wolfdieter Lang, Oct 18 2004

Keywords

Comments

((-1)^(n+1))*a(n) = S_{-11}(n), n>=0, defined in A092184.

Links

Programs

  • Mathematica
    LinearRecurrence[{12,12,-1},{0,1,11},30] (* Harvey P. Dale, Mar 20 2023 *)

Formula

a(n)= 2*(T(n, 13/2)-(-1)^n)/15, with twice Chebyshev's polynomials of the first kind evaluated at x=13/2: 2*T(n, 13/2)=A078363(n)=((13+sqrt(165))^n + (13-sqrt(165))^n)/2^n.
a(n)= 13*a(n-1)-a(n-2)+2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.
a(n)= 12*a(n-1) + 12*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=11.
G.f.: x*(1-x)/((1+x)*(1-13*x+x^2)) = x*(1-x)/(1-12*x-12*x^2+x^3) (from the Stephan link, see A092184).

A099271 Unsigned member r=-13 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 13, 196, 2925, 43681, 652288, 9740641, 145457325, 2172119236, 32436331213, 484372848961, 7233156403200, 108012973199041, 1612961441582413, 24086408650537156, 359683168316474925, 5371161116096586721
Offset: 0

Views

Author

Wolfdieter Lang, Oct 18 2004

Keywords

Comments

((-1)^(n+1))*a(n) = S_{-13}(n), n>=0, defined in A092184.

Links

Programs

  • Mathematica
    LinearRecurrence[{14,14,-1},{0,1,13},41] (* or *) CoefficientList[Series[ (x-x^2)/(1-14 x-14 x^2+x^3),{x,0,40}],x] (* Harvey P. Dale, Jun 18 2011 *)

Formula

a(n)= 2*(T(n, 15/2)-(-1)^n)/17, with twice Chebyshev's polynomials of the first kind evaluated at x=15/2: 2*T(n, 15/2)=A078365(n)=((15+sqrt(221))^n + (15-sqrt(221))^n)/2^n.
a(n)= 15*a(n-1)-a(n-2)+2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.
a(n)= 14*a(n-1) + 14*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=13.
G.f.: x*(1-x)/((1+x)*(1-15*x+x^2)) = x*(1-x)/(1-14*x-14*x^2+x^3) (from the Stephan link, see A092184).

A099273 Unsigned member r=-15 of the family of Chebyshev sequences S_r(n) defined in A092184.

Original entry on oeis.org

0, 1, 15, 256, 4335, 73441, 1244160, 21077281, 357069615, 6049106176, 102477735375, 1736072395201, 29410752983040, 498246728316481, 8440783628397135, 142995074954434816, 2422475490596994735
Offset: 0

Views

Author

Wolfdieter Lang, Oct 18 2004

Keywords

Comments

((-1)^(n+1))*a(n) = S_{-15}(n), n>=0, defined in A092184.

Links

Programs

  • Mathematica
    LinearRecurrence[{16,16,-1},{0,1,15},30] (* Harvey P. Dale, Oct 09 2011 *)

Formula

a(n)= 2*(T(n, 17/2)-(-1)^n)/19, with twice Chebyshev's polynomials of the first kind evaluated at x=17/2: 2*T(n, 17/2)=A078367(n)= ((17+sqrt(285))^n +(17-sqrt(285))^n)/2^n.
a(n)= 17*a(n-1)-a(n-2)+2*(-1)^(n+1), n>=2, a(0)=0, a(1)=1.
a(n)= 16*a(n-1) + 16*a(n-2) - a(n-3), n>=3, a(0)=0, a(1)=1, a(2)=15.
G.f.: x*(1-x)/((1+x)*(1-17*x+x^2)) = x*(1-x)/(1-16*x-16*x^2+x^3) (from the Stephan link, see A092184).
Previous Showing 11-20 of 46 results. Next

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