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 1-3 of 3 results.

A098244 First differences of Chebyshev polynomials S(n,171)=A097844(n) with Diophantine property.

Original entry on oeis.org

1, 170, 29069, 4970629, 849948490, 145336221161, 24851643870041, 4249485765555850, 726637214266180309, 124250714153751276989, 21246145483077202184810, 3632966626892047822325521, 621216047053057100415479281, 106224311079445872123224631530
Offset: 0

Views

Author

Wolfdieter Lang, Sep 10 2004

Keywords

Comments

(13*b(n))^2 - 173*a(n)^2 = -4 with b(n)=A097845(n) give all positive solutions of this Pell equation.

Examples

			All positive solutions of Pell equation x^2 - 173*y^2 = -4 are (13=13*1,1), (2236=13*172,170), (382343=13*29411,29069), (65378417=13*5029109,4970629), ...

Links

Programs

  • GAP
    a:=[1,170];; for n in [3..20] do a[n]:=171*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
  • Magma
    I:=[1,170]; [n le 2 select I[n] else 171*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
  • Mathematica
    LinearRecurrence[{171,-1}, {1,170}, 20] (* G. C. Greubel, Aug 01 2019 *)
  • PARI
    my(x='x+O('x^20)); Vec((1-x)/(1-171*x+x^2)) \\ G. C. Greubel, Aug 01 2019
  • Sage
    ((1-x)/(1-171*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

Formula

a(n) = ((-1)^n)*S(2*n, 13*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1-171*x+x^2).
a(n) = S(n, 171) - S(n-1, 171) = T(2*n+1, sqrt(173)/2)/(sqrt(173)/2), with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x) = 0 = U(-1, x) and T(n, x) Chebyshev's polynomials of the first kind, A053120.
a(n) = 171*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=170. - Philippe Deléham, Nov 18 2008

A140455 13-Fibonacci sequence.

Original entry on oeis.org

0, 1, 13, 170, 2223, 29069, 380120, 4970629, 64998297, 849948490, 11114328667, 145336221161, 1900485203760, 24851643870041, 324971855514293, 4249485765555850, 55568286807740343, 726637214266180309
Offset: 0

Views

Author

R. J. Mathar, Jul 22 2008

Keywords

Comments

The k-Fibonacci sequences for k=2..12 are A000129, A006190, A001076, A052918, A005668, A054413, A041025, A099371, A041041, A049666, A041061. This here is k=13. k=14 is A041085, k=16 A041113, k=18 A041145, k=20 A041181, k=22 A041221.
For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010
For n>=2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 13's along the main diagonal and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
For n>=1, a(n) equals the number of words of length n-1 on alphabet {0,1,...,13} avoiding runs of zeros of odd length. - Milan Janjic, Jan 28 2015
From Michael A. Allen, Apr 21 2023: (Start)
Also called the 13-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 13 kinds of squares available. (End)

Links

Crossrefs

Row n=13 of A073133, A172236 and A352361 and column k=13 of A157103.

Programs

  • Maple
    F := proc(n,k) coeftayl( x/(1-k*x-x^2),x=0,n) ; end: for n from 0 to 20 do printf("%d,",F(n,13)) ; od:
  • Mathematica
    LinearRecurrence[{13, 1}, {0, 1}, 30] (* Vincenzo Librandi, Nov 17 2012 *)
  • Sage
    [lucas_number1(n,13,-1) for n in range(0, 18)] # Zerinvary Lajos, Apr 29 2009

Formula

O.g.f.: x/(1-13*x-x^2).
a(n) = 13*a(n-1) + a(n-2).
a(n-r)*a(n+r) - a(n)^2 = (-1)^(n+1-r)*a(r)^2.
a(n) = Sum_{i=0..floor((n-1)/2)} binomial(n,2i+1)*13^(n-1-2*i)*(13^2+4)^i/2^(n-1).
a(n) = ((13+sqrt(173))^n - (13-sqrt(173))^n)/(2^n*sqrt(173)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2*n) = 13*A097844(n), a(2*n+1) = A098244(n).
a(3*n+1) = A041319(5*n), a(3*n+2) = A041319(5*n+3), a(3*n+3) = 2*A041319(5*n+4).
Limit_{k->oo} a(n+k)/a(k) = (A088316(n) + A140455(n)*sqrt(173))/2.
Limit_{n->oo} A088316(n)/A140455(n) = sqrt(173). (End)

A097845 Chebyshev polynomials S(n,171) + S(n-1,171) with Diophantine property.

Original entry on oeis.org

1, 172, 29411, 5029109, 859948228, 147046117879, 25144026209081, 4299481435634972, 735186181467371131, 125712537549484828429, 21496108734780438290228, 3675708881109905462800559
Offset: 0

Views

Author

Wolfdieter Lang, Sep 10 2004

Keywords

Comments

(13*a(n))^2 - 173*b(n)^2 = -4 with b(n) = A098244(n) give all positive solutions of this Pell equation.

Examples

			All positive solutions of Pell equation x^2 - 173*y^2 = -4 are (13 = 13*1,1), (2236 = 13*172,170), (382343 = 13*29411,29069), (65378417 = 13*5029109,4970629), ...

Links

Crossrefs

Cf. A049310, A098244.

Programs

  • GAP
    a:=[1,172];; for n in [3..20] do a[n]:=171*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 14 2019
  • Magma
    m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)/(1-171*x+x^2) )); // G. C. Greubel, Jan 14 2019
  • Mathematica
    LinearRecurrence[{171,-1},{1,172},20] (* Harvey P. Dale, Feb 27 2012 *)
CoefficientList[Series[(1+x)/(1-171*x+x^2), {x, 0, 20}], x] (* Stefano Spezia, Jan 14 2019 *)
  • PARI
    Vec((1+x)/(1-171*x+x^2)+O(x^20)) \\ Charles R Greathouse IV, Feb 08 2017
  • Sage
    ((1+x)/(1-171*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 14 2019

Formula

a(n) = S(n, 171) + S(n-1, 171) = S(2*n, sqrt(173)), with S(n, x) = U(n, x/2) Chebyshev's polynomials of the second kind, A049310. S(-1, x) = 0 = U(-1, x). S(n, 171) = A097844(n).
a(n) = (-2/13)*i*((-1)^n)*T(2*n+1, 13*i/2) with the imaginary unit i and Chebyshev's polynomials of the first kind. See the T-triangle A053120.
G.f.: (1+x)/(1-171*x+x^2).
a(n) = 171*a(n-1) - a(n-2), n > 1, a(0)=1, a(1)=172. - Philippe Deléham, Nov 18 2008
Showing 1-3 of 3 results.

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