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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.

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A049072 Expansion of 1/(1 - 3*x + 4*x^2).

Original entry on oeis.org

1, 3, 5, 3, -11, -45, -91, -93, 85, 627, 1541, 2115, 181, -7917, -24475, -41757, -27371, 84915, 364229, 753027, 802165, -605613, -5025499, -12654045, -17860139, -2964237, 62547845, 199500483, 348310069, 246928275, -652455451, -2945079453, -6225416555
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

From Sharon Sela (sharonsela(AT)hotmail.com), Jan 22 2002: (Start)
a(n) is the determinant of the following tridiagonal n X n matrix:
[3 2 0 0 .... ]
[2 3 2 0 .... ]
[0 2 3 2 0 .. ]
[. 0 2 3 2 .. ]
[. . . . .... ]
[. . . 2 3 2 0]
[. . . 0 2 3 2]
[. . . 0 0 2 3]
(End)
With offset 1 (a(0) = 0, a(1) = 1) this is a divisibility sequence. - R. K. Guy, May 19 2015
With offset 1 (a(0) = 0, a(1) = 1), then this is the Lucas sequence U_n(P, Q) = U_n(3, 4). V_n(P, Q) = V_n(3, 4) = A247563(n). Again with offset 1 (a(0) = 0, a(1) = 1), then (A247563(n)/2)^2 + 7(a(n)/2)^2 = 4^n. This is a specific case of the Lucas sequence identity (V_n/2)^2 - D*(U_n/2)^2 = Q^n where V_n = (a^n + b^n), U_n = (a^n - b^n)/(a - b), Q = (a*b) = 4 and D = (a - b)^2 = -7; a = (3 + sqrt(-7))/2 and b = (3 - sqrt(-7))/2. - Raphie Frank, Dec 04 2015

Examples

			G.f.: 1 + 3*x + 5*x^2 + 3*x^3 - 11*x^4 - 45*x^5 - 91*x^6 - 93*x^7 + ...

Links

Crossrefs

Cf. A006516, A025170, A247563.

Programs

  • Haskell
    a049072 n = a049072_list !! n
a049072_list = 1 : 3 :
    zipWith (-) (map (* 3) $ tail a049072_list) (map (* 4) a049072_list)
-- Reinhard Zumkeller, Oct 25 2013
  • Magma
    I:=[1,3]; [n le 2 select I[n] else 3*Self(n-1)-4*Self(n-2): n in [1..50]]; // Vincenzo Librandi, Jun 12 2015
  • Maple
    A049072:=n->(-1)^n*add(binomial(2*n-k+1,k)*(-2)^k, k=0..n): seq(A049072(n), n=0..40); # Wesley Ivan Hurt, Dec 05 2015
  • Mathematica
    Join[{a=1,b=3},Table[c=3*b-4*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
a[ n_] := ChebyshevU[ n, 3/4] 2^n; (* Michael Somos, Jun 03 2015 *)
a[ n_] := Module[ {m = n + 1, s = 1}, If[ m < 0, {m, s} = -{m, 4^m}]; s SeriesCoefficient[ x / (1 - 3 x + 4 x^2), {x, 0, m}]]; (* Michael Somos, Jun 03 2015 *)
  • PARI
    {a(n) = 2^n * subst( -3*poltchebi(n+1) + 4*poltchebi(n), 'x, 3/4) * 4/7}; /* Michael Somos, Sep 15 2005 */
  • PARI
    {a(n) = if(n<0, 0, matdet(matrix(n, n, i, j, if(abs(i-j)<2, 3-abs(i-j)))))} /* Michael Somos, Sep 15 2005 */
  • PARI
    {a(n) = polchebyshev(n, 2, 3/4) * 2^n}; /* Michael Somos, Jun 03 2015 */
  • PARI
    x='x+O('x^100); Vec(1/(1-3*x+4*x^2)) \\ Altug Alkan, Dec 04 2015
  • Sage
    [lucas_number1(n,3,4) for n in range(1, 34)] # Zerinvary Lajos, Apr 23 2009

Formula

G.f.: 1/(1 - 3*x + 4*x^2).
a(n) = (-1)^n * Sum_{k=0..n} binomial(2n-k+1, k)*(-2)^k. - Paul Barry, Jan 17 2005
a(n) = 3*a(n-1) - 4*a(n-2); a(0)=1, a(1)=3. - Sergei N. Gladkovskii, Mar 14 2013
G.f.: 1/(1/Q(0)+2*x^3) where Q(k) = 1 + k*(2*x+1) + 8*x - 2*x*(k+1)*(k+5)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(n) = - a(-2-n) * 4^(n+1) for all n in Z. - Michael Somos, Jun 03 2015
a(n - 1) = (((3 + sqrt(-7))/2)^n - ((3 - sqrt(-7))/2)^n)/(((3 + sqrt(-7))/2) - ((3 - sqrt(-7))/2)). - Raphie Frank, Dec 04 2015

A247564 a(n) = 3*a(n-2) - 4*a(n-4) with a(0) = 2, a(1) = 1, a(2) = 3, a(3) = 1.

Original entry on oeis.org

2, 1, 3, 1, 1, -1, -9, -7, -31, -17, -57, -23, -47, -1, 87, 89, 449, 271, 999, 457, 1201, 287, -393, -967, -5983, -4049, -16377, -8279, -25199, -8641, -10089, 7193, 70529, 56143, 251943, 139657, 473713, 194399, 413367, 24569, -654751, -703889, -3617721
Offset: 0

Views

Author

Michael Somos, Sep 20 2014

Keywords

Examples

			G.f. = 2 + x + 3*x^2 + x^3 + x^4 - x^5 - 9*x^6 - 7*x^7 - 31*x^8 - 17*x^9 + ...

Links

Crossrefs

Cf. A247487, A247560, A247563.

Programs

  • Haskell
    a247564 n = a247564_list !! n
a247564_list = [2,1,3,1] ++ zipWith (-) (map (* 3) $ drop 2 a247564_list)
                                        (map (* 4) $ a247564_list)
-- Reinhard Zumkeller, Sep 20 2014
  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((2+x-3*x^2-2*x^3)/(1-3*x^2+4*x^4)));  // G. C. Greubel, Aug 04 2018
  • Maple
    H := (n, a, b) -> hypergeom([a - n/2, b - n/2], [1 - n], 8):
a := n -> `if`(n < 3, [2, 1, 3][n+1], (-1)^iquo(n, 2)*H(n, irem(n, 2), 1/2)):
seq(simplify(a(n)), n=0..42); # Peter Luschny, Sep 03 2019
# second Maple program:
a:= n-> (<<0|1>, <-4|3>>^iquo(n, 2, 'r').<[<2, 3>, <1, 1>][1+r]>)[1,1]:
seq(a(n), n=0..42);  # Alois P. Heinz, Sep 03 2019
  • Mathematica
    CoefficientList[Series[(2+x-3*x^2-2*x^3)/(1-3*x^2+4*x^4), {x,0,60}], x] (* G. C. Greubel, Aug 04 2018 *)
  • PARI
    {a(n) = if( n<0, n=-n; 2^-n, 1) * polcoeff( (2 + x - 3*x^2 - 2*x^3) / (1 - 3*x^2 + 4*x^4) + x * O(x^n), n)};

Formula

G.f.: (2 + x - 3*x^2 - 2*x^3) / (1 - 3*x^2 + 4*x^4).
a(n) = A247487(n) * 3^( n == 1 (mod 4) ) for all n in Z.
a(2*n) = A247563(n). a(2*n + 1) = A247560(n).
0 = a(n)*(+2*a(n+2)) + a(n+1)*(+2*a(n+1) - 8*a(n+2) + a(n+3)) + a(n+2)*(+a(n+2)) for all n in Z.
a(n) = (-1)^floor(n/2)*H(n, n mod 2, 1/2) for n >= 3 where H(n, a, b) = hypergeom([a - n/2, b - n/2], [1 - n], 8). - Peter Luschny, Sep 03 2019
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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