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

A137878 Perfect squares among 17-gonal numbers A051869(k) = k*(15*k - 13)/2.

Original entry on oeis.org

1, 17689, 378225, 4109707449, 87870152041, 954775454112481, 20414169462254569, 221815343046210267025, 4742660677722035990769, 51532584126226886201833161, 1101824413949324675985344641, 11972153009151467313136073526409, 255978051492792346696545201859225
Offset: 1

Views

Author

Alexander Adamchuk, Feb 19 2008

Keywords

Comments

Corresponding square roots sqrt(a(n)) are listed in A137879.
Indices of perfect squares among the 17-gonal numbers A051869(k) = k*(15*k - 13)/2 are listed in A137880. Note that all such indices are also perfect squares, their square roots are listed in A137881(k) = sqrt(A137880(k)).

Links

Crossrefs

Cf. A051869 (17-gonal numbers), A137879, A137880, A137881.

Programs

  • PARI
    Vec(x*(1+17688*x+128214*x^2+17688*x^3+x^4)/((1-x)*(1-482*x+x^2)*(1+482*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 19 2016

Formula

a(n) = A137879(n)^2 = A051869( A137880(n) ) = A051869( A137881(n)^2 ).
From Colin Barker, Jun 19 2016: (Start)
a(n) = a(n-1) + 232322*a(n-2) - 232322*a(n-3) - a(n-4) + a(n-5) for n > 5.
G.f.: x*(1 + 17688*x + 128214*x^2 + 17688*x^3 + x^4) / ((1-x)*(1 - 482*x + x^2)*(1 + 482*x + x^2)).
(End)

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
Showing 1-3 of 3 results.

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