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 1-5 of 5 results.

A077410 Combined Diophantine Chebyshev sequences A077249 and A077251.

Original entry on oeis.org

1, 2, 12, 21, 119, 208, 1178, 2059, 11661, 20382, 115432, 201761, 1142659, 1997228, 11311158, 19770519, 111968921, 195707962, 1108378052, 1937309101, 10971811599, 19177383048, 108609737938
Offset: 0

Views

Author

Wolfdieter Lang, Nov 08 2002

Keywords

Comments

-24*a(n)^2 + b(n)^2 = 25, with the companion sequence b(n)= A077411(n).

Examples

			24*a(2)^2 + 25 = 24*12^2 + 25 = 3481 = 59^2 = A077411(2)^2.

Links

Programs

  • Magma
    I:=[1,2,12,21]; [n le 4 select I[n] else 10*Self(n-2) - Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 18 2018
  • Mathematica
    CoefficientList[Series[(1+x)*(1+x+x^2)/(1-10*x^2+x^4), {x,0,50}], x] (* or *) LinearRecurrence[{0,10,0,-1}, {1,2,12,21}, 30] (* G. C. Greubel, Jan 18 2018 *)
  • PARI
    x='x+O('x^30); Vec((1+x)*(1+x+x^2)/(1-10*x^2+x^4)) \\ G. C. Greubel, Jan 18 2018

Formula

a(2*k) = A077251(k) and a(2*k+1) = A077249(k), k>=0.
a(n) = sqrt((A077411(n)^2 - 25)/24).
G.f.: (1+x)*(1+x+x^2)/(1-10*x^2+x^4).

A077250 Bisection (odd part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

11, 103, 1019, 10087, 99851, 988423, 9784379, 96855367, 958769291, 9490837543, 93949606139, 930005223847, 9206102632331, 91131021099463, 902104108362299, 8929910062523527, 88396996516872971, 875040055106206183, 8662003554545188859, 85744995490345682407
Offset: 0

Views

Author

Wolfdieter Lang, Nov 08 2002

Keywords

Comments

a(n)^2 - 24*b(n)^2 = 25, with the companion sequence b(n) = A077249(n).
The even part is A077409(n) with Diophantine companion A077251(n).

Examples

			103 = a(1) = sqrt(24*A077249(1)^2 + 25) = sqrt(24*21^2 + 25) = sqrt(10609) = 103.

Links

Programs

Formula

a(n) = 10*a(n-1)- a(n-2), a(-1)=7, a(0)=11.
a(n) = 2*T(n+1, 5)+T(n, 5), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 5)= A001079(n).
a(n) = sqrt(25 + 24*A077249(n)^2).
G.f.: (11-7*x)/(1-10*x+x^2).

A077251 Bisection (even part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

1, 12, 119, 1178, 11661, 115432, 1142659, 11311158, 111968921, 1108378052, 10971811599, 108609737938, 1075125567781, 10642645939872, 105351333830939, 1042870692369518, 10323355589864241, 102190685206272892, 1011583496472864679, 10013644279522373898
Offset: 0

Views

Author

Wolfdieter Lang, Nov 08 2002

Keywords

Comments

b(n)^2 - 24*a(n)^2 = 25, with the companion sequence b(n) = A077409(n).
The odd part is A077249(n) with Diophantine companion A077250(n).

Examples

			24*a(1)^2 + 25 = 24*12^2 + 25 = 3481 = 59^2 = A077409(1)^2.

Links

Programs

Formula

a(n) = 10*a(n-1)- a(n-2), a(-1)=-2, a(0)=1.
a(n) = S(n, 10)+2*S(n-1, 10), with S(n, x) = U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310. S(n, 10)= A004189(n+1).
a(n) = sqrt((A077409(n)^2 - 25)/24).
G.f.: (1+2*x)/(1-10*x+x^2).

A077409 Bisection (even part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

7, 59, 583, 5771, 57127, 565499, 5597863, 55413131, 548533447, 5429921339, 53750679943, 532076878091, 5267018100967, 52138104131579, 516114023214823, 5109002128016651, 50573907256951687, 500630070441500219, 4955726797158050503, 49056637901139004811
Offset: 0

Views

Author

Wolfdieter Lang, Nov 08 2002

Keywords

Comments

a(n)^2 - 24*b(n)^2 = 25, with the companion sequence b(n) = A077251(n).
The odd part is A077250(n) with Diophantine companion A077249(n).

Examples

			59 = a(1) = sqrt(24*A077251(1)^2 + 25) = sqrt(24*12^2 + 25) = sqrt(3481) = 59.

Links

Programs

  • Magma
    I:=[7,59]; [n le 2 select I[n] else 10*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 18 2018
  • Mathematica
    CoefficientList[Series[(7 - 11 z)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{10,-1}, {7,59}, 30] (* G. C. Greubel, Jan 18 2018 *)
  • PARI
    a(n)=if(n<0,0,subst(poltchebi(n+1)+2*poltchebi(n),x,5))
  • PARI
    Vec((7-11*x)/(1-10*x+x^2) + O(x^30)) \\ Colin Barker, Jun 15 2015
  • PARI
    a(n)=polchebyshev(n+1,,5)+2*polchebyshev(n,,5) \\ Charles R Greathouse IV, Jun 15 2015
  • PARI
    a(n)=([0,1;-1,10]^n*[7;59])[1,1] \\ Charles R Greathouse IV, Jun 15 2015

Formula

a(n) = 10*a(n-1)- a(n-2), a(-1)=11, a(0)=7.
a(n) = T(n+1, 5)+2*T(n, 5), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 5) = A001079(n).
a(n) = sqrt(24*A077251(n)^2 + 25).
G.f.: (7-11*x)/(1-10*x+x^2).

A106331 Numbers j such that 24*(j^2) + 25 = k^2.

Original entry on oeis.org

0, 1, 2, 5, 12, 21, 50, 119, 208, 495, 1178, 2059, 4900, 11661, 20382, 48505, 115432, 201761, 480150, 1142659, 1997228, 4752995, 11311158, 19770519, 47049800, 111968921, 195707962, 465745005, 1108378052, 1937309101, 4610400250
Offset: 1

Views

Author

Pierre CAMI, Apr 29 2005

Keywords

Comments

The ratio k(n) /(2*j(n)) tends to sqrt(6) as n increases

Links

Crossrefs

Cf. A106330.

Formula

j(1)=0, j(2)=1, j(3)=2, j(4)=5, j(5)=10*j(2)+j(3), j(6)=10*j(3)+j(2), j(7)=10*j(4)+j(1) then j(n)=10*j(n-3)-j(n-6).
a(n) = +10*a(n-3) -a(n-6). G.f.: x^2*(1+2*x+5*x^2+2*x^3+x^4)/(1-10*x^3+x^6). [R. J. Mathar, May 22 2010]
a(3*n+1) = 5*A004189(n), a(3*n+2) = A077251(n), a(3*n+3) = A077249(n). - Ralf Stephan, Nov 15 2010

Extensions

More terms from Jon E. Schoenfield, May 16 2010
Showing 1-5 of 5 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.