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.

A048907 Indices of 9-gonal numbers which are also triangular.

Original entry on oeis.org

1, 10, 154, 2449, 39025, 621946, 9912106, 157971745, 2517635809, 40124201194, 639469583290, 10191389131441, 162422756519761, 2588572715184730, 41254740686435914, 657487278267789889, 10478541711598202305, 166999180107303446986, 2661508340005256949466
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

Entries are == 1 (mod 3). - N. J. A. Sloane, Sep 22 2007
lim(n -> Infinity, a(n)/a(n-1)) = 8 + 3*sqrt(7). - Ant King, Nov 03 2011

Links

Crossrefs

Cf. A001080, A073352, A048908, A048909.

Programs

  • Mathematica
    LinearRecurrence[{17, -17, 1}, {1, 10, 154}, 17]; (* Ant King, Nov 03 2011 *)
  • PARI
    Vec(-x*(x^2-7*x+1)/((x-1)*(x^2-16*x+1)) + O(x^20)) \\ Colin Barker, Jun 22 2015

Formula

G.f.: x*(1-7*x+x^2)/((1-x)*(1-16*x+x^2)).
a(n+2) = 16*a(n+1)-a(n)-5, a(n+1) = 8*a(n)-2.5+1.5*(28*a(n)^2-20*a(n)+1)^0.5. - Richard Choulet, Sep 22 2007
From Ant King, Nov 03 2011: (Start)
a(n) = 17*a(n-1) - 17*a(n-2) + a(n-3).
a(n) = ceiling(3/28*(3-sqrt(7))*(8 + 3*sqrt(7))^n).
(End)
a(n) = A097830(n-1)-7*A097830(n-2)+A097830(n-3). - R. J. Mathar, Jul 04 2024

A048908 Indices of triangular numbers which are also 9-gonal.

Original entry on oeis.org

1, 25, 406, 6478, 103249, 1645513, 26224966, 417953950, 6661038241, 106158657913, 1691877488374, 26963881156078, 429730221008881, 6848719654986025, 109149784258767526, 1739547828485294398, 27723615471505942849, 441838299715609791193, 7041689179978250716246
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Comments

lim( n -> Infinity , a(n)/a(n-1)) = 8 + 3*sqrt(7). - Ant King, Nov 03 2011

Links

Crossrefs

Cf. A048907, A048909.

Programs

  • Mathematica
    LinearRecurrence[{17, -17, 1}, {1, 25, 406}, 16]; (* Ant King, Nov 03 2011 *)
  • PARI
    Vec(x*(2*x^2-8*x-1)/((x-1)*(x^2-16*x+1)) + O(x^50)) \\ Colin Barker, Jun 22 2015

Formula

a(n+2) = 16*a(n+1)-a(n)+7, a(n+1) = 8*a(n)+3.5+1.5*(28*a(n)^2+28*a(n)+25)^0.5 - Richard Choulet, Sep 22 2007
G.f.: f(z) = a(1)*z+a(2)*z^2+... = (z+8z^2-2*z^3)/((1-z)*(1-16*z+z^2)) - Richard Choulet, Oct 09 2007
From Ant King, Nov 03 2011: (Start)
a(n) = 17*a(n-1) - 17*a(n-2) + a(n-3).
a(n) = floor(3/28*sqrt(7)*(3 - sqrt(7))*(8 + 3* sqrt(7))^n).
(End)

A128922 Numbers simultaneously 10-gonal and centered 10-gonal.

Original entry on oeis.org

1, 451, 145351, 46802701, 15070324501, 4852597686751, 1562521384809451, 503127033310956601, 162005342204743216201, 52165217062894004660251, 16797037888909664757384751
Offset: 0

Views

Author

Steven Schlicker, Apr 24 2007

Keywords

Examples

			a(1) = 451 because 451 is the tenth centered 10-gonal number and the eleventh 10-gonal number.

Links

Crossrefs

Cf. A001107, A062786, A048909.

Programs

  • Maple
    CP := n -> 1+1/2*10*(n^2-n): N:=10: u:=9: v:=1: x:=10: y:=1: k_pcp:=[1]: for i from 1 to N do tempx:=x; tempy:=y; x:=tempx*u+80*tempy*v: y:=tempx*v+tempy*u: s:=(y+1)/2: k_pcp:=[op(k_pcp),CP(s)]: end do: k_pcp;

Formula

Let x(n) + y(n)*sqrt(80) =: (10+sqrt(80))*(9+sqrt(80))^n, s(n) = (y(n)+1)/2; then a(n) = (1/2)*(2+10*(s(n)^2-s(n))).
From Richard Choulet, Oct 01 2007: (Start)
a(n+2) = 322*a(n+1)-a(n)+130.
a(n+1) = 161*a(n)+65+9*(320*a(n)^2+260*a(n)+45)^0.5.
G.f.: z*(1+128*z+z^2)/((1-z)*(1-322*z+z^2)). (End)
Showing 1-3 of 3 results.

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

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