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.

A097841 First differences of Chebyshev polynomials S(n,83) = A097839(n) with Diophantine property.

Original entry on oeis.org

1, 82, 6805, 564733, 46866034, 3889316089, 322766369353, 26785719340210, 2222891938868077, 184473245206710181, 15309056460218076946, 1270467212952893676337, 105433469618629957059025
Offset: 0

Views

Author

Wolfdieter Lang, Sep 10 2004

Keywords

Comments

(9*b(n))^2 - 85*a(n)^2 = -4 with b(n)=A097840(n) give all positive solutions of this Pell equation.
For n > 0, a(n) is the hypotenuse of the Pythagorean triple (x(n), y(n), a(n)) that is primitive for n == 0, 2 (mod 3) where (x(n)) and (y(n)) are recurrences of the form (82,82,-1) with the initial values x(0) = 1, x(1) = 80, x(2) = 6643; y(0) = 0, y(1) = 18, y(2) = 1476. - Klaus Purath, Jul 19 2025

Examples

			All positive solutions of Pell equation x^2 - 85*y^2 = -4 are (9=9*1,1), (756=9*84,82), (62739=9*6971,6805), (5206581=9*578509,564733), ...

Links

Crossrefs

Cf. A097839, A097840.
Cf. A049310, A053120.
Cf. A099371.

Programs

  • GAP
    a:=[1,82];; for n in [3..20] do a[n]:=83*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 13 2019
  • Magma
    m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)/(1-83*x+x^2) )); // G. C. Greubel, Jan 13 2019
  • Mathematica
    CoefficientList[Series[(1-x)/(1-83x+x^2), {x, 0, 20}], x] (* Michael De Vlieger, Feb 08 2017 *)
LinearRecurrence[{83,-1}, {1,82}, 20] (* G. C. Greubel, Jan 13 2019 *)
  • PARI
    my(x='x+O('x^20)); Vec((1-x)/(1-83*x+x^2)) \\ G. C. Greubel, Jan 13 2019
  • Sage
    ((1-x)/(1-83*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 13 2019

Formula

a(n) = ((-1)^n)*S(2*n, 9*i) with the imaginary unit i and the S(n, x) = U(n, x/2) Chebyshev polynomials.
G.f.: (1-x)/(1 - 83*x + x^2).
a(n) = S(n, 83) - S(n-1, 83) = T(2*n+1, sqrt(85)/2)/(sqrt(85)/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) = 83*a(n-1) - a(n-2) for n > 1, a(0)=1, a(1)=82. - Philippe Deléham, Nov 18 2008
From Klaus Purath, Jul 19 2025: (Start)
a(n) = A099371(2n+1) = A099371(n)^2 + A099371(n+1)^2.
a(n) = (t(i+2*n+1) + t(i))/(t(i+n+1) + t(i+n)) as long as t(i+n+1) + t(i+n) != 0 for any integer i and n >= 1 where (t) is a sequence satisfying t(i+3) = 82*t(i+2) + 82*t(i+1) - t(i) or t(i+2) = 83*t(i+1) - t(i) regardless of initial values and including this sequence itself. (End)

A099371 Expansion of g.f.: x/(1 - 9*x - x^2).

Original entry on oeis.org

0, 1, 9, 82, 747, 6805, 61992, 564733, 5144589, 46866034, 426938895, 3889316089, 35430783696, 322766369353, 2940328107873, 26785719340210, 244011802169763, 2222891938868077, 20250039251982456, 184473245206710181, 1680509246112374085, 15309056460218076946
Offset: 0

Views

Author

Wolfdieter Lang, Oct 18 2004

Keywords

Comments

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 9'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,...,9} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Michael A. Allen, Mar 10 2023: (Start)
Also called the 9-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 9 kinds of squares available. (End)

Links

Crossrefs

Row n=9 of A073133, A172236 and A352361.
Cf. A099372 (squares).
Cf. A041151, A054413, A086902, A087798, A099371, A097839, A178765, A243399.

Programs

  • Magma
    I:=[0,1]; [n le 2 select I[n] else 9*Self(n-1) + Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018
  • Maple
    F:= gfun:-rectoproc({a(n)=9*a(n-1)+a(n-2),a(0)=0,a(1)=1},a(n),remember):
seq(F(n),n=0..30); # Robert Israel, Feb 01 2015
  • Mathematica
    CoefficientList[Series[x/(1-9*x-x^2), {x,0,30}], x] (* G. C. Greubel, Apr 16 2017 *)
LinearRecurrence[{9,1}, {0,1}, 30] (* G. C. Greubel, Jan 24 2018 *)
  • PARI
    my(x='x+O('x^30)); concat([0], Vec(1/(1-9*x-x^2)) ) \\ Charles R Greathouse IV, Feb 03 2014
  • Sage
    from sage.combinat.sloane_functions import recur_gen3
it = recur_gen3(0,1,9,9,1,0)
[next(it) for i in range(1,22)] # Zerinvary Lajos, Jul 09 2008
  • Sage
    [lucas_number1(n,9,-1) for n in range(0, 20)] # Zerinvary Lajos, Apr 26 2009

Formula

G.f.: x/(1 - 9*x - x^2).
a(n) = 9*a(n-1) + a(n-2), n >= 2, a(0)=0, a(1)=1.
a(n) = (-i)^(n-1)*S(n-1, 9*i) with S(n, x) Chebyshev's polynomials of the second kind (see A049310) and i^2=-1.
a(n) = (ap^n - am^p)/(ap-am) with ap:= (9+sqrt(85))/2 and am:= (9-sqrt(85))/2 = -1/ap (Binet form).
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1-k, k)*9^(n-1-2*k) n >= 1.
a(n) = F(n, 9), the n-th Fibonacci polynomial evaluated at x=9. - T. D. Noe, Jan 19 2006
a(n) = ((9+sqrt(85))^n - (9-sqrt(85))^n)/(2^n*sqrt(85)). Offset 1. a(3)=82. - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009
a(p) == 85^((p-1)/2) (mod p) for odd primes p. - Gary W. Adamson, Feb 22 2009 [See A087475 for more info about this congruence. - Jason Yuen, Apr 05 2025]
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+2) = 9*A097839(n), a(2n+1) = A097841(n).
a(3n+1) = A041151(5n), a(3n+2) = A041151(5n+3), a(3n+3) = 2*A041151(5n+4).
Limit_{k -> infinity} (a(n+k)/a(k)) = (A087798(n) + A099371(n)*sqrt(85))/2.
Lim_{n->infinity} A087798(n)/A099371(n) = sqrt(85). (End)
a(n) ~ 1/sqrt(85)*((9+sqrt(85))/2)^n. - Jean-François Alcover, Dec 04 2013
a(n) = [1,0] (M^n) [0,1]^T where M is the matrix [9,1; 1,0]. - Robert Israel, Feb 01 2015
E.g.f.: 2*exp(9*x/2)*sinh(sqrt(85)*x/2)/sqrt(85). - Stefano Spezia, Apr 06 2023

A097840 Chebyshev polynomials S(n,83) + S(n-1,83) with Diophantine property.

Original entry on oeis.org

1, 84, 6971, 578509, 48009276, 3984191399, 330639876841, 27439125586404, 2277116783794691, 188973253929372949, 15682502959354160076, 1301458772372465913359, 108005395603955316648721
Offset: 0

Views

Author

Wolfdieter Lang, Sep 10 2004

Keywords

Comments

(9*a(n))^2 - 85*b(n)^2 = -4 with b(n)=A097841(n) give all positive solutions of this Pell equation.

Examples

			All positive solutions of Pell equation x^2 - 85*y^2 = -4 are (9=9*1,1), (756=9*84,82), (62739=9*6971,6805), (5206581=9*578509,564733), ...

Links

Crossrefs

Cf. A097783, A097834, A097837, A097839.

Programs

  • GAP
    a:=[1,84];; for n in [3..20] do a[n]:=83*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Jan 13 2019
  • Magma
    m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)/(1-83*x+x^2) )); // G. C. Greubel, Jan 13 2019
  • Mathematica
    CoefficientList[Series[(1+x)/(1-83x+x^2), {x, 0, 20}], x] (* Michael De Vlieger, Feb 08 2017 *)
  • PARI
    my(x='x+O('x^20)); Vec((1+x)/(1-83*x+x^2)) \\ G. C. Greubel, Jan 13 2019
  • Sage
    ((1+x)/(1-83*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 13 2019

Formula

a(n) = S(n, 83) + S(n-1, 83) = S(2*n, sqrt(85)), 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, 83) = A097839(n).
a(n) = (-2/9)*i*((-1)^n)*T(2*n+1, 9*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 - 83*x + x^2).
a(n) = 83*a(n-1) - a(n-2) for n > 1; a(0)=1, a(1)=84. - Philippe Deléham, Nov 18 2008
From Peter Bala, Aug 26 2022: (Start)
a(n) = (2/9)*(9/2 o 9/2 o ... o 9/2) (2*n+1 terms), where the binary operation o is defined on real numbers by x o y = x*sqrt(1 + y^2) + y*sqrt(1 + x^2). The operation o is commutative and associative with identity 0.
The aerated sequence (b(n))n>=1 = [1, 0, 84, 0, 6971, 0, 578509, 0, ...], with o.g.f. x*(1 + x^2)/(1 - 83*x^2 + x^4), is a fourth-order linear divisibility sequence; that is, if n | m then b(n) | b(m). It is the case P1 = 0, P2 = -81, Q = -1 of the 3-parameter family of divisibility sequences found by Williams and Guy. See A100047 for the connection with Chebyshev polynomials.
b(n) = 1/2*( (-1)^n - 1 )*F(n,9) + 1/9*( 1 + (-1)^(n+1) )*F(n+1,9), where F(n,x) is the n-th Fibonacci polynomial - see A168561 (but with row indexing starting at n = 1).
Exp( Sum_{n >= 1} 18*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 18*A099371(n)*x^n.
Exp( Sum_{n >= 1} (-18)*b(n)*x^n/n ) = 1 + Sum_{n >= 1} 18*A099371(n)*(-x)^n. (End)

A099373 Twice Chebyshev polynomials of the first kind, T(n,x), evaluated at 83/2.

Original entry on oeis.org

2, 83, 6887, 571538, 47430767, 3936182123, 326655685442, 27108485709563, 2249677658208287, 186696137145578258, 15493529705424787127, 1285776269413111753283, 106703936831582850735362, 8855140980751963499281763, 734869997465581387589650967, 60985354648662503206441748498
Offset: 0

Views

Author

Wolfdieter Lang, Oct 18 2004

Keywords

Comments

Used in A099372.
The proper and improper nonnegative solutions of the Pell equation x(n)^2 - 85*y(n)^2 = +4 are x(n) = a(n) and y(n) = 9*A097839(n), n >= 0. - Wolfdieter Lang, Jul 01 2013

Examples

			Pell equation: n=0: 2^2 - 85*0^2 = +4 (improper), n=1: 83^2 - 85*(9*1)^2 = +4, n=2: 6887^2 - 85*(9*83)^2 = +4. - _Wolfdieter Lang_, Jul 01 2013

Links

Crossrefs

Cf. A049310, A053120, A097839, A099372.

Programs

  • Mathematica
    LinearRecurrence[{83,-1},{2,83},20] (* Harvey P. Dale, Apr 07 2025 *)

Formula

a(n) = 83*a(n-1) - a(n-2), n >= 1; a(-1) = 83, a(0) = 2.
a(n) = S(n, 83) - S(n-2, 83) = 2*T(n, 83/2) with S(n, x) := U(n, x/2), S(-1, x) := 0, S(-2, x) := -1. S(n, 83) = A097839(n). U-, resp. T-, are Chebyshev polynomials of the second, resp. first, case. See A049310 and A053120.
G.f.: (2 - 83*x)/(1 - 83*x + x^2).
a(n) = ap^n + am^n, with ap := (83+9*sqrt(85))/2 and am := (83-9*sqrt(85))/2.
E.g.f.: 2*exp(83*x/2)*cosh(9*sqrt(85)*x/2). - Stefano Spezia, Apr 06 2023
Showing 1-4 of 4 results.

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