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.

A077248 Combined Diophantine Chebyshev sequences A077246 and A077244.

Original entry on oeis.org

2, 3, 13, 22, 102, 173, 803, 1362, 6322, 10723, 49773, 84422, 391862, 664653, 3085123, 5232802, 24289122, 41197763, 191227853, 324349302, 1505533702, 2553596653, 11853041763, 20104423922, 93318800402
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)= A077247(n).
Positive values of x (or y) satisfying x^2 - 8xy + y^2 + 35 = 0. - Colin Barker, Feb 08 2014

Examples

			13 = a(2) = sqrt((5*A077247(2)^2 + 7)/3) = sqrt((5*10^2 + 7)/3)= sqrt(169) = 13.

Links

Programs

  • Mathematica
    CoefficientList[Series[(1 - x) (2 + x) (1 + 2 x)/(1 - 8 x^2 + x^4), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 11 2014 *)

Formula

a(2*k)= A077246(k) and a(2*k+1)= A077244(k), k>=0.
G.f.: (1-x)*(2+x)*(1+2*x)/(1-8*x^2+x^4).

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).

A077245 Bisection (even part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

1, 10, 79, 622, 4897, 38554, 303535, 2389726, 18814273, 148124458, 1166181391, 9181326670, 72284431969, 569094129082, 4480468600687, 35274654676414, 277716768810625, 2186459495808586, 17213959197658063
Offset: 0

Views

Author

Wolfdieter Lang, Nov 08 2002

Keywords

Comments

3*b(n)^2 - 5*a(n)^2 = 7, with the companion sequence b(n)= A077246(n).
The odd part is A077243(n) with Diophantine companion A077244(n).

Examples

			5*a(1)^2 + 7 = 5*10^2 + 7 = 507 = 3*13^2 = 3*A077246(1)^2.

Links

Formula

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