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.

A077242 Combined Diophantine Chebyshev sequences A077240 and A077239.

Original entry on oeis.org

5, 7, 23, 37, 133, 215, 775, 1253, 4517, 7303, 26327, 42565, 153445, 248087, 894343, 1445957, 5212613, 8427655, 30381335, 49119973, 177075397, 286292183, 1032071047, 1668633125, 6015350885, 9725506567, 35060034263, 56684406277, 204344854693, 330380931095
Offset: 0

Views

Author

Wolfdieter Lang, Nov 08 2002

Keywords

Comments

a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A077241(n).
Because there is only one primitive Pythagorean triangle with sum of the legs L = 17 (see also A120681), namely (5,12,13), all positive solutions (x(n), y(n)) = (a(n), 2*A077241(n)) of the (generalized) Pell equation x^2 - 2*y^2 = +17 satisfy x(n) < 2*y(n), for n >= 1, only 5 = x(0) > 2*y(0) = 4. The proof runs along the same line as the one given in a comment on the L=7 case in A077443. - Wolfdieter Lang, Feb 05 2015

Examples

			23 = a(2) = sqrt(8*A077241(2)^2 + 17) = sqrt(8*8^2 + 17)= sqrt(529) = 23.

Links

Programs

  • Magma
    I:=[5,7,23,37]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Feb 18 2014
  • Mathematica
    A077239 = Table[2*ChebyshevT[n+1, 3] + ChebyshevT[n, 3], {n, 0, 12}]; A077240 = Table[ChebyshevT[n+1, 3] + 2*ChebyshevT[n, 3], {n, 0, 12}]; Riffle[A077240, A077239] (* Jean-François Alcover, Dec 19 2013 *)
CoefficientList[Series[(1 - x) (5 + 12 x + 5 x^2)/(1 - 6 x^2 + x^4), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 18 2014 *)
  • PARI
    Vec((1-x)*(5+12*x+5*x^2)/(1-6*x^2+x^4) + O(x^50)) \\ Colin Barker, Mar 27 2016

Formula

a(2*k) = A077240(k) and a(2*k+1) = A077239(k), k>=0.
G.f.: (1-x)*(5+12*x+5*x^2)/(1-6*x^2+x^4).
a(n) = 6*a(n-2)-a(n-4) for n>3. - Vincenzo Librandi, Feb 18 2014
a(n) = ((6-5*sqrt(2))*(1-sqrt(2))^n - (-1-sqrt(2))^n*(-4+sqrt(2)) + 4*(-1+sqrt(2))^n + sqrt(2)*(-1+sqrt(2))^n + 6*(1+sqrt(2))^n + 5*sqrt(2)*(1+sqrt(2))^n)/4. - Colin Barker, Mar 27 2016

A054488 Expansion of (1+2*x)/(1-6*x+x^2).

Original entry on oeis.org

1, 8, 47, 274, 1597, 9308, 54251, 316198, 1842937, 10741424, 62605607, 364892218, 2126747701, 12395593988, 72246816227, 421085303374, 2454265004017, 14304504720728, 83372763320351, 485932075201378, 2832219687887917
Offset: 0

Views

Author

Barry E. Williams, May 04 2000

Keywords

Comments

Bisection (even part) of Chebyshev sequence with Diophantine property.
b(n)^2 - 8*a(n)^2 = 17, with the companion sequence b(n)= A077240(n).
The odd part is A077413(n) with Diophantine companion A077239(n).

Examples

			8 = a(1) = sqrt((A077240(1)^2 - 17)/8) = sqrt((23^2 - 17)/8)= sqrt(64) = 8.

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N. Y., 1964, pp. 122-125, 194-196.

Links

Crossrefs

Cf. A000129, A002203, A002315, A038761, A077239, A077240, A077413.
Cf. A077241 (even and odd parts).

Programs

  • GAP
    a:=[1,8];; for n in [3..30] do a[n]:=6*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 19 2020
  • Magma
    I:=[1,8]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 19 2020
  • Magma
    R:=PowerSeriesRing(Integers(), 21); Coefficients(R!( (1+2*x)/(1-6*x+x^2))); // Marius A. Burtea, Jan 20 2020
  • Maple
    a[0]:=1: a[1]:=8: for n from 2 to 26 do a[n]:=6*a[n-1]-a[n-2] od: seq(a[n], n=0..20); # Zerinvary Lajos, Jul 26 2006
  • Mathematica
    LinearRecurrence[{6,-1},{1,8},30] (* Harvey P. Dale, Oct 09 2017 *)
Table[(LucasL[2*n+1, 2] + Fibonacci[2*n, 2])/2, {n,0,30}] (* G. C. Greubel, Jan 19 2020 *)
  • PARI
    my(x='x+O('x^30)); Vec((1+2*x)/(1-6*x+x^2)) \\ G. C. Greubel, Jan 19 2020
  • PARI
    apply( {A054488(n)=[1,8]*([0,-1;1,6]^n)[,1]}, [0..30]) \\ M. F. Hasler, Feb 27 2020
  • Sage
    [(lucas_number2(2*n+1,2,-1) + lucas_number1(2*n,2,-1))/2 for n in (0..30)] # G. C. Greubel, Jan 19 2020

Formula

a(n) = 6*a(n-1) - a(n-2), a(0)=1, a(1)=8.
a(n) = ((3 + 2*sqrt(2))^(n+1) - (3 - 2*sqrt(2))^(n+1) + 2*((3 + 2*sqrt(2))^n - (3 - 2*sqrt(2))^n))/(4*sqrt(2)).
a(n) = S(n, 6) + 2*S(n-1, 6), with S(n, x) Chebyshev's polynomials of the second kind, A049310. S(n, 6) = A001109(n+1).
a(n) = (-1)^n*Sum_{k = 0..n} A238731(n,k)*(-9)^k. - Philippe Deléham, Mar 05 2014
a(n) = (Pell(2*n+2) + 2*Pell(2*n))/2 = (Pell-Lucas(2*n+1) + Pell(2*n))/2. - G. C. Greubel, Jan 19 2020
E.g.f.: (1/4)*exp(3*x)*(4*cosh(2*sqrt(2)*x) + 5*sqrt(2)*sinh(2*sqrt(2)*x)). - Stefano Spezia, Jan 27 2020

Extensions

More terms from James Sellers, May 05 2000
Chebyshev comments from Wolfdieter Lang, Nov 08 2002

A077413 Bisection (odd part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

2, 13, 76, 443, 2582, 15049, 87712, 511223, 2979626, 17366533, 101219572, 589950899, 3438485822, 20040964033, 116807298376, 680802826223, 3968009658962, 23127255127549, 134795521106332, 785645871510443, 4579079707956326, 26688832376227513, 155553914549408752
Offset: 0

Views

Author

Wolfdieter Lang, Nov 08 2002

Keywords

Comments

-8*a(n)^2 + b(n)^2 = 17, with the companion sequence b(n) = A077239(n).
The even part is A054488(n) with Diophantine companion A077240(n).

Examples

			8*a(1)^2 + 17 = 8*13^2+17 = 1369 = 37^2 = A077239(1)^2.
G.f. = 2 + 13*x + 76*x^2 + 443*x^3 + 2582*x^4 + ... - _Michael Somos_, Jul 30 2024

Links

Crossrefs

Cf. A077241 (even and odd parts), A001109, A054488.

Programs

  • Magma
    I:=[2,13]; [n le 2 select I[n] else 6*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 18 2018
  • Mathematica
    LinearRecurrence[{6,-1}, {2,13}, 30] (* or *) CoefficientList[Series[ (2+x)/(1-6*x+x^2), {x, 0, 50}], x] (* G. C. Greubel, Jan 18 2018 *)
a[ n_] := 2*ChebyshevU[n, 3] + ChebyshevU[n-1, 3]; (* Michael Somos, Jul 30 2024 *)
  • PARI
    Vec((2+x)/(1-6*x+x^2) + O(x^30)) \\ Colin Barker, Jun 16 2015
  • PARI
    {a(n) = 2*polchebyshev(n, 2, 3) + polchebyshev(n-1, 2, 3)}; /* Michael Somos, Jul 30 2024 */
  • PARI
    {a(n) = my(w = 3 + quadgen(32)); imag(w^n + 2*w^(n+1))}; /* Michael Somos, Jul 30 2024 */

Formula

a(n) = 6*a(n-1) - a(n-2), a(-1)=-1, a(0)=2.
a(n) = 2*S(n, 6)+S(n-1, 6), with S(n, x) = U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(n, 6) = A001109(n+1).
G.f.: (2+x)/(1-6*x+x^2).
a(n) = (((3-2*sqrt(2))^n*(-7+4*sqrt(2))+(3+2*sqrt(2))^n*(7+4*sqrt(2))))/(4*sqrt(2)). - Colin Barker, Oct 12 2015
a(n) = -A054488(-1-n) for all n in Z. - Michael Somos, Jul 30 2024

A077239 Bisection (odd part) of Chebyshev sequence with Diophantine property.

Original entry on oeis.org

7, 37, 215, 1253, 7303, 42565, 248087, 1445957, 8427655, 49119973, 286292183, 1668633125, 9725506567, 56684406277, 330380931095, 1925601180293, 11223226150663, 65413755723685, 381259308191447, 2222142093424997, 12951593252358535, 75487417420726213
Offset: 0

Views

Author

Wolfdieter Lang, Nov 08 2002

Keywords

Comments

a(n)^2 - 8*b(n)^2 = 17, with the companion sequence b(n)= A077413(n).
The even part is A077240(n) with Diophantine companion A054488(n).

Examples

			37 = a(1) = sqrt(8*A077413(1)^2 +17) = sqrt(8*13^2 + 17)= sqrt(1369) = 37.

Links

Crossrefs

Cf. A077242 (even and odd parts).

Programs

  • Mathematica
    Table[2*ChebyshevT[n+1, 3] + ChebyshevT[n, 3], {n, 0, 19}]  (* Jean-François Alcover, Dec 19 2013 *)
  • PARI
    Vec((7-5*x)/(1-6*x+x^2) + O(x^40)) \\ Colin Barker, Oct 12 2015

Formula

a(n) = 6*a(n-1) - a(n-2), a(-1) := 5, a(0)=7.
a(n) = 2*T(n+1, 3)+T(n, 3), with T(n, x) Chebyshev's polynomials of the first kind, A053120. T(n, 3)= A001541(n).
G.f.: (7-5*x)/(1-6*x+x^2).
a(n) = (((3-2*sqrt(2))^n*(-8+7*sqrt(2))+(3+2*sqrt(2))^n*(8+7*sqrt(2))))/(2*sqrt(2)). - Colin Barker, Oct 12 2015
Showing 1-4 of 4 results.

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