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.

Showing 1-6 of 6 results.

A154021 a(n+2) = 16*a(n+1) - a(n), with a(1)=0, a(2)=4.

Original entry on oeis.org

0, 4, 64, 1020, 16256, 259076, 4128960, 65804284, 1048739584, 16714029060, 266375725376, 4245297576956, 67658385505920, 1078288870517764, 17184963542778304, 273881127813935100, 4364913081480183296
Offset: 1

Views

Author

Vincenzo Librandi, Jan 04 2009

Keywords

Comments

If a(n)=x and a(n+1)=y, then 16=(x^2+y^2)/(xy+1).
In general, the sequence a(1)=0, a(2)=U; a(n+2)=U^2*a(n+1)-a(n) has the property that "If a(n)=x and a(n+1)=y then (x^2+y^2)/(xy+1)=U^2".

Links

Crossrefs

Cf. A065100, A154022-A154027.

Programs

  • Magma
    I:=[0,4]; [n le 2 select I[n] else 16*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Feb 25 2012
  • Mathematica
    Nest[Append[#,16Last[#]-#[[-2]]]&,{0,4},20]  (* or *) Rest[CoefficientList[Series[4x^2/(1-16x+x^2), {x,0,22}], x]]  (* Harvey P. Dale, Apr 17 2011 *)
LinearRecurrence[{16, -1}, {0, 4}, 20] (* T. D. Noe, Apr 17 2011 *)

Formula

From R. J. Mathar, Jan 05 2011: (Start)
G.f.: 4*x^2/(1 -16*x +x^2).
a(n) = 4*A077412(n-2). (End)

Extensions

375725376 replaced by 266375725376 - R. J. Mathar, Jan 07 2009
Edited by N. J. A. Sloane, Jun 23 2010 at the suggestion of Joerg Arndt.

A154022 a(n) = 5*A097780(n-2).

Original entry on oeis.org

0, 5, 125, 3120, 77875, 1943755, 48516000, 1210956245, 30225390125, 754423796880, 18830369531875, 470004814499995, 11731289992968000, 292812245009700005, 7308574835249532125, 182421558636228603120
Offset: 1

Views

Author

Vincenzo Librandi, Jan 04 2009

Keywords

Comments

If a(n)=x and a(n+1)=y, then (x^2+y^2)/(xy+1)=25.

Links

Crossrefs

Cf. A065100, A154021-A154027.

Programs

  • Magma
    I:=[0, 5]; [n le 2 select I[n] else 25*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jan 29 2012
  • Mathematica
    CoefficientList[Series[(5*z)/(z^2-25*z+1),{z,0,25}],z] (* Vincenzo Librandi, Jan 29 2012 *)
LinearRecurrence[{25,-1},{0,5},20] (* Harvey P. Dale, Mar 06 2018 *)
  • PARI
    concat(0,Vec(5/(1-25*x+x^2)+O(x^98))) \\ Charles R Greathouse IV, Dec 27 2011

Formula

a(n) = +25*a(n-1) -a(n-2).
G.f.: 5*x^2/(1 -25*x +x^2). - R. J. Mathar, Jan 05 2011

Extensions

Edited by N. J. A. Sloane, Jun 23 2010 at the suggestion of Joerg Arndt.

A154026 a(n+2) = 81*a(n+1) - a(n), a(1)=0, a(2)=9.

Original entry on oeis.org

0, 9, 729, 59040, 4781511, 387243351, 31361929920, 2539929080169, 205702893563769, 16659394449585120, 1349205247522830951, 109268965654899721911, 8849437012799354643840, 716695129071092826429129, 58043456017745719586115609, 4700803242308332193648935200
Offset: 1

Views

Author

Vincenzo Librandi, Jan 04 2009

Keywords

Comments

If a(n)=x and a(n+1)=y then (x^2+y^2)/(xy+1)=81.

Links

Crossrefs

Cf. A065100, A154021-A154027.

Programs

  • Magma
    I:=[0,9]; [n le 2 select I[n] else 81*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 25 2012
  • Mathematica
    LinearRecurrence[{81,-1},{0,9},20] (* Harvey P. Dale, Sep 15 2011 *)
  • PARI
    concat(0,Vec(9/(x^2-81*x+1)+O(x^98))) \\ Charles R Greathouse IV, Dec 27 2011

Formula

G.f.: (9*x)/(1 -81*x +x^2). - Harvey P. Dale, Sep 15 2011

Extensions

More terms from Harvey P. Dale, Sep 15 2011

A154023 a(n+2) = 36*a(n+1) - a(n), a(1)=0, a(2)=6.

Original entry on oeis.org

0, 6, 216, 7770, 279504, 10054374, 361677960, 13010352186, 468011000736, 16835385674310, 605605873274424, 21784976052204954, 783653532006103920, 28189742176167536166, 1014047064810025198056, 36477504590984739593850
Offset: 1

Views

Author

Vincenzo Librandi, Jan 04 2009

Keywords

Comments

If a(n)=x and a(n+1)=y then (x^2+y^2)/(xy+1)=36.

Links

Crossrefs

Cf. A065100, A154021-A154027.

Programs

Formula

From R. J. Mathar, Oct 18 2010: (Start)
a(n)= +36*a(n-1) -a(n-2)
a(n) = 6*A144128(n-1).
G.f.: 6*x/(1 -36*x +x^2). (End)

Extensions

Edited by N. J. A. Sloane, Jun 23 2010 at the suggestion of Joerg Arndt.
Missing digit inserted in a(8) by R. J. Mathar, Oct 18 2010

A154024 a(n+2) = 49*a(n+1) - a(n), a(1)=0, a(2)=7.

Original entry on oeis.org

0, 7, 343, 16800, 822857, 40303193, 1974033600, 96687343207, 4735705783543, 231952896050400, 11360956200686057, 556454900937566393, 27254929189740067200, 1334935075396325726407
Offset: 1

Views

Author

Vincenzo Librandi, Jan 04 2009

Keywords

Comments

If a(n)=x and a(n+1)=y, then (x^2+y^2)/(xy+1)=49.

Links

Crossrefs

Cf. A065100, A154021-A154027.

Programs

  • Magma
    I:=[0, 7]; [n le 2 select I[n] else 49*Self(n-1)-Self(n-2): n in [1..50]]; // Vincenzo Librandi, Feb 25 2012
  • Mathematica
    LinearRecurrence[{49,-1},{0,7},30] (* Vincenzo Librandi, Feb 25 2012 *)
Rest@ CoefficientList[Series[7 x^2/(1 - 49 x + x^2), {x, 0, 14}], x] (* Michael De Vlieger, Sep 01 2016 *)
  • PARI
    concat(0,Vec(7/(1-49*x+x^2)+O(x^98))) \\ Charles R Greathouse IV, Dec 27 2011

Formula

G.f.: 7*x^2/(1 -49*x +x^2). - R. J. Mathar, Jan 05 2011

A154025 a(n+2) = 64*a(n+1) - a(n), a(1)=0, a(2)=8.

Original entry on oeis.org

0, 8, 512, 32760, 2096128, 134119432, 8581547520, 549084921848, 35132853450752, 2247953535926280, 143833893445831168, 9203121226997268472, 588855924634379351040, 37677576055373281198088, 2410776011619255617326592
Offset: 1

Views

Author

Vincenzo Librandi, Jan 04 2009

Keywords

Comments

If a(n)=x and a(n+1)=y then (x^2+y^2)/(xy+1)=64.

Links

Crossrefs

Cf. A065100, A154021-A154027.

Programs

  • Magma
    I:=[0, 8]; [n le 2 select I[n] else 64*Self(n-1)-Self(n-2): n in [1..20]]; // Vincenzo Librandi, Jan 29 2012
  • Mathematica
    CoefficientList[Series[(8z)/(z^2-64z+1),{z,0,20}],z] (* Vincenzo Librandi, Jan 29 2012 *)
LinearRecurrence[{64,-1},{0,8},20] (* Harvey P. Dale, Dec 19 2023 *)
  • PARI
    a(n)=([0,1; -1,64]^(n-1)*[0;8])[1,1] \\ Charles R Greathouse IV, Sep 01 2016

Formula

G.f.: 8*x^2/(1 -64*x +x^2). - R. J. Mathar, Jan 05 2011
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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