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.

A077411 Combined Diophantine Chebyshev sequences A077409 and A077250.

Original entry on oeis.org

7, 11, 59, 103, 583, 1019, 5771, 10087, 57127, 99851, 565499, 988423, 5597863, 9784379, 55413131, 96855367, 548533447, 958769291, 5429921339, 9490837543, 53750679943, 93949606139, 532076878091, 930005223847, 5267018100967
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)= A077410(n).

Examples

			59 = a(2) = sqrt(24*A077410(2)^2 + 25) = sqrt(24*12^2 + 25)= sqrt(3481) = 59.

Links

Programs

  • Magma
    I:=[7,11,59,103]; [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)*(7+18*x+7*x^2)/(1-10*x^2+x^4), {x,0,50}], x] (* or *) LinearRecurrence[{0,10,0,-1}, {7,11,59,103}, 30] (* G. C. Greubel, Jan 18 2018 *)
  • PARI
    x='x+O('x^30); Vec((1-x)*(7+18*x+7*x^2)/(1-10*x^2+x^4)) \\ G. C. Greubel, Jan 18 2018

Formula

a(2*k)= A077409(k) and a(2*k+1)= A077250(k), k>=0.
a(n)= sqrt(24*A077410(n)^2 + 25).
G.f.: (1-x)*(7+18*x+7*x^2)/(1-10*x^2+x^4).

A077249 Bisection (odd part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

2, 21, 208, 2059, 20382, 201761, 1997228, 19770519, 195707962, 1937309101, 19177383048, 189836521379, 1879187830742, 18602041786041, 184141230029668, 1822810258510639, 18043961355076722, 178616803292256581, 1768124071567489088, 17502623912382634299
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) = A077250(n).
The even part is A077251(n) with Diophantine companion A077409(n).

Examples

			24*a(1)^2 + 25 = 24*21^2+25 = 10609 = 103^2 = A077250(1)^2.

Links

Programs

  • Mathematica
    CoefficientList[Series[(z + 2)/(z^2 - 10 z + 1), {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 11 2011 *)
LinearRecurrence[{10,-1},{2,21},40] (* Harvey P. Dale, Apr 08 2012 *)
  • PARI
    a(n)=if(n<0,0,subst(-7*poltchebi(n)+11*poltchebi(n+1),x,5)/24)
  • PARI
    a(n)=2*polchebyshev(n,2,5)+polchebyshev(n-1,2,5) \\ Charles R Greathouse IV, Jun 11 2011
  • PARI
    Vec((2+x)/(1-10*x+x^2) + O(x^30)) \\ Colin Barker, Jun 15 2015

Formula

a(n) = 10*a(n-1)- a(n-2), a(-1) := -1, a(0)=2.
a(n) = 2*S(n, 10)+S(n-1, 10), with S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 10)= A004189(n+1).
G.f.: (2+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).

A106330 Numbers k such that k^2 = 24*j^2 + 25.

Original entry on oeis.org

5, 7, 11, 25, 59, 103, 245, 583, 1019, 2425, 5771, 10087, 24005, 57127, 99851, 237625, 565499, 988423, 2352245, 5597863, 9784379, 23284825, 55413131, 96855367, 230496005, 548533447, 958769291, 2281675225, 5429921339, 9490837543, 22586256245, 53750679943
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.
k(n) = 2*b + 1, for n > 0, where b is a side of the Heronian triangle (5, b, b+1). - Andrés Ventas, Dec 13 2024

Links

Crossrefs

Cf. A106331.

Programs

  • PARI
    Vec(-x*(7*x^5+11*x^4+25*x^3-11*x^2-7*x-5)/(x^6-10*x^3+1) + O(x^100)) \\ Colin Barker, Apr 16 2014

Formula

Recurrence: k(1)=5, k(2)=7, k(3)=11, k(4)=25, k(5)=10*k(2)-k(3), k(6)=10*k(3)-k(2) then k(n)=10*k(n-3)-k(n-6).
From Ralf Stephan, Nov 15 2010: (Start)
G.f.: (-7x^5-11x^4-25x^3+11x^2+7x+5)/(x^6-10x^3+1).
a(3n+1) = 5*A001079(n), a(3n+2) = A077409(n), a(3n+3) = A077250(n). (End)

Extensions

More terms from Ralf Stephan, Nov 15 2010
More terms from Colin Barker, Apr 16 2014
Showing 1-5 of 5 results.

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