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 1-3 of 3 results.

A137881 a(n) = sqrt(A137880(n)).

Original entry on oeis.org

1, 7, 15, 153, 329, 3359, 7223, 73745, 158577, 1619031, 3481471, 35544937, 76433785, 780369583, 1678061799, 17132585889, 36840925793, 376136519975, 808822305647, 8257870853561, 17757249798441, 181297022258367, 389850673260055, 3980276618830513, 8558957561922769
Offset: 1

Views

Author

Alexander Adamchuk, Feb 19 2008

Keywords

Comments

A137880 gives the indices m (= a(n)^2) of perfect squares in 17-gonal numbers A051869(m) = m(15m -13)/2. Corresponding 17-gonal numbers are listed in A137878(n) = A051869( a(n)^2 ).
Positive values of x (or y) satisfying x^2 - 22xy + y^2 + 104 = 0. - Colin Barker, Feb 19 2014

Links

Crossrefs

Cf. A051869 (17-gonal numbers), A137878 (17-gonal numbers that are perfect squares), A137879, A137880.

Programs

  • Magma
    I:=[1,7,15,153]; [n le 4 select I[n] else 22*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Feb 21 2014
  • Mathematica
    CoefficientList[Series[(1 - x) (x^2 + 8 x + 1)/(x^4 - 22 x^2 + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 21 2014 *)

Formula

a(n) = sqrt(A137880(n)). A051869( a(n)^2 ) = A137878(n).
For n>=5, a(n) = 22*a(n-2) - a(n-4). [Alekseyev]
a(2n) = (15 - sqrt(30))/30 * (11 + 2*sqrt(30))^n + (15 + sqrt(30))/30 * (11 - 2*sqrt(30))^n. [Alekseyev]
a(2n+1) = (15 + sqrt(30))/30 * (11 + 2*sqrt(30))^n + (15 - sqrt(30))/30 * (11 - 2*sqrt(30))^n. [Alekseyev]
G.f.: -x*(x-1)*(x^2+8*x+1) / (x^4-22*x^2+1). - Colin Barker, Feb 19 2014

Extensions

Edited and extended by Max Alekseyev, Oct 19 2008

A137879 Numbers k such that k^2 is a 17-gonal number.

Original entry on oeis.org

1, 133, 615, 64107, 296429, 30899441, 142878163, 14893466455, 68866978137, 7178619931869, 33193740583871, 3460079913694403, 15999314094447685, 1667751339780770377, 7711636199783200299, 803852685694417627311, 3716992648981408096433
Offset: 1

Views

Author

Alexander Adamchuk, Feb 19 2008

Keywords

Comments

Corresponding 17-gonal numbers equal k^2 are listed in A137878.
The 17-gonal numbers A051869(n) = n*(15n - 13)/2 are perfect squares for indices n listed in A137880. Note that all such indices are also perfect squares of numbers listed in A137881.

Links

Crossrefs

Cf. A051869 (17-gonal numbers), A137878 (17-gonal numbers that are perfect squares), A137880, A137881.

Programs

Formula

a(n) = sqrt(A137878(n)) = sqrt(A051869(A137880(n))) = sqrt(A051869(A137881(n)^2)).
From Max Alekseyev, Oct 19 2008: (Start)
For n>=5, a(n) = 482*a(n-2) - a(n-4).
a(2n) = (-60 + 17*sqrt(30))/120 * (11 + 2*sqrt(30))^(2n) + (-60 - 17*sqrt(30))/120 * (11 - 2*sqrt(30))^(2n).
a(2n+1) = (60 + 17*sqrt(30))/120 * (11 + 2*sqrt(30))^(2n) + (60 - 17*sqrt(30))/120 * (11 - 2*sqrt(30))^(2n). (End)

Extensions

Extended by Max Alekseyev, Oct 19 2008

A137880 Indices k of perfect squares among 17-gonal numbers A051869(k) = k*(15*k - 13)/2.

Original entry on oeis.org

1, 49, 225, 23409, 108241, 11282881, 52171729, 5438325025, 25146664929, 2621261378961, 12120640323841, 1263442546333969, 5842123489426225, 608976686071593889, 2815891401263116401, 293525499243961920321, 1357253813285332678849, 141478681658903574000625
Offset: 1

Views

Author

Alexander Adamchuk, Feb 19 2008

Keywords

Comments

Corresponding perfect squares are listed in A137878.
Note that all a(n) are perfect squares themselves, their square roots are listed in A137881.

Links

Crossrefs

Cf. A051869 (17-gonal numbers), A137878 (17-gonal numbers that are perfect squares), A137879, A137881.

Programs

  • Mathematica
    Rest@ CoefficientList[Series[x (1 + 48 x - 306 x^2 + 48 x^3 + x^4)/((1 - x) (1 - 22 x + x^2) (1 + 22 x + x^2)), {x, 0, 18}], x] (* Michael De Vlieger, Jun 18 2016 *)
  • PARI
    Vec(x*(1+48*x-306*x^2+48*x^3+x^4)/((1-x)*(1-22*x+x^2)*(1+22*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 18 2016

Formula

A051869( a(n) ) = A137878(n); a(n) = A137881(n)^2.
From Max Alekseyev, Oct 19 2008: (Start)
a(n) = 482*a(n-2) - a(n-4) - 208.
a(2n) = ( (15 - sqrt(30))/30 * (11 + 2*sqrt(30))^n + (15 + sqrt(30))/30 * (11 - 2*sqrt(30))^n )^2.
a(2n+1) = ( (15 + sqrt(30))/30 * (11 + 2*sqrt(30))^n + (15 - sqrt(30))/30 * (11 - 2*sqrt(30))^n )^2. (End)
a(n) = a(n-1) + 482*a(n-2) - 482*a(n-3) - a(n-4) + a(n-5). - Matthew House, Jun 18 2016
G.f.: x*(1 + 48*x - 306*x^2 + 48*x^3 + x^4) / ((1-x)*(1 - 22*x + x^2)*(1 + 22*x + x^2)). - Colin Barker, Jun 18 2016

Extensions

Edited and extended by Max Alekseyev, Oct 19 2008
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.