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 1-4 of 4 results.

A077247 Combined Diophantine Chebyshev sequences A077245 and A077243.

Original entry on oeis.org

1, 2, 10, 17, 79, 134, 622, 1055, 4897, 8306, 38554, 65393, 303535, 514838, 2389726, 4053311, 18814273, 31911650, 148124458, 251239889, 1166181391, 1978007462, 9181326670, 15572819807, 72284431969, 122604550994, 569094129082
Offset: 0

Views

Author

Wolfdieter Lang, Nov 08 2002

Keywords

Comments

-5*a(n)^2 + 3*b(n)^2 = 7, with the companion sequence b(n)= A077248(n).
In addition to the comment above: 3*b(n)^2 = 5*a(n-2)*a(n+2) + 112, where b(n) = (a(n+2) - a(n-2))/6 = A077248(n), n >= 2. - Klaus Purath, Aug 12 2021

Examples

			5*a(1)^2 + 7 = 5*4 + 7 = 27 = 3*3^2 = 3*A077248(1)^2.

Links

Crossrefs

Cf. A077243, A077245, A077248.

Programs

  • Mathematica
    LinearRecurrence[{0,8,0,-1},{1,2,10,17},30] (* Harvey P. Dale, Nov 12 2022 *)

Formula

a(2*k)= A077245(k) and a(2*k+1)= A077243(k), k>=0.
G.f.: (1+x)*(1+x+x^2)/(1-8*x^2+x^4).
From Klaus Purath, Aug 12 2021: (Start)
a(n) = 8*a(n-2) - a(n-4), n >= 4.
a(n) = (a(n-2)*a(n-4) - 168)/a(n-6), n >= 6.
a(n) = (a(n-1)*a(n-2) - 15/2 - 9/2*(-1)^n)/a(n-3), n >= 3. (End)

A077244 Bisection (odd part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

3, 22, 173, 1362, 10723, 84422, 664653, 5232802, 41197763, 324349302, 2553596653, 20104423922, 158281794723, 1246149933862, 9810917676173, 77241191475522, 608118614128003, 4787707721548502, 37693543158260013, 296760637544531602, 2336391557197992803
Offset: 0

Views

Author

Wolfdieter Lang, Nov 08 2002

Keywords

Comments

3*a(n)^2 - 5*b(n)^2 = 7, with the companion sequence b(n)= A077243(n).
The even part is A077246(n) with Diophantine companion A077245(n).

Examples

			22 = a(1) = sqrt((5*A077243(1)^2 + 7)/3) = sqrt((5*17^2 + 7)/3) = sqrt(484) = 22.

Links

Programs

  • Magma
    I:=[3,22]; [n le 2 select I[n] else 8*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Oct 12 2015
  • Mathematica
     LinearRecurrence[{8, -1}, {3, 22}, 25] (* Vincenzo Librandi, Oct 12 2015 *)
  • PARI
    Vec((3-2*x)/(1-8*x+x^2) + O(x^40)) \\ Colin Barker, Oct 12 2015

Formula

a(n)= (2*T(n+1, 4)+T(n, 4))/3, with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 4)= A001091(n).
G.f.: (3-2*x)/(1-8*x+x^2).
From Colin Barker, Oct 12 2015: (Start)
a(n) = (((4-sqrt(15))^n * (-10+3*sqrt(15)) + (4+sqrt(15))^n * (10+3*sqrt(15)))) / (2*sqrt(15)).
a(n) = 8*a(n-1) - a(n-2).
(End)

A077243 Bisection (odd part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

2, 17, 134, 1055, 8306, 65393, 514838, 4053311, 31911650, 251239889, 1978007462, 15572819807, 122604550994, 965263588145, 7599504154166, 59830769645183, 471046653007298, 3708542454413201, 29197292982298310
Offset: 0

Views

Author

Wolfdieter Lang, Nov 08 2002

Keywords

Comments

-5*a(n)^2 + 3* b(n)^2 = 7, with the companion sequence b(n)= A077244(n).
The even part is A077245(n) with Diophantine companion A077246(n).

Examples

			5*a(1)^2 + 7 = 5*17^2+7 = 1452 = 3*22^2 = 3*A077244(1)^2.

Links

Programs

  • Mathematica
    LinearRecurrence[{8,-1},{2,17},30] (* Harvey P. Dale, Oct 03 2015 *)

Formula

a(n)= 8*a(n-1) - a(n-2), a(-1)=-1, a(0)=2.
a(n)= 2*S(n, 8)+S(n-1, 8), with S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 8)= A001090(n+1).
G.f.: (2+x)/(1-8*x+x^2).

A077246 Bisection (even part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

2, 13, 102, 803, 6322, 49773, 391862, 3085123, 24289122, 191227853, 1505533702, 11853041763, 93318800402, 734697361453, 5784260091222, 45539383368323, 358530806855362, 2822707071474573, 22223125764941222
Offset: 0

Views

Author

Wolfdieter Lang, Nov 08 2002

Keywords

Comments

3*a(n)^2 - 5*b(n)^2 = 7, with the companion sequence b(n)= A077245(n).
The odd part is A077244(n) with Diophantine companion A077243(n).

Examples

			13 = a(1) = sqrt((5*A077245(1)^2 + 7)/3) = sqrt((5*10^2 + 7)/3) = sqrt(169) = 13.

Links

Programs

  • Mathematica
    LinearRecurrence[{8,-1},{2,13},30] (* Harvey P. Dale, Apr 30 2012 *)

Formula

a(n)= 8*a(n-1) - a(n-2), a(-1) := 3, a(0)=2.
a(n)= (T(n+1, 4)+2*T(n, 4))/3, with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 4)= A001091(n).
G.f.: (2-3*x)/(1-8*x+x^2).
Showing 1-4 of 4 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.