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

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

0,2

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

B. R. Myers, On Spanning Trees, Weighted Compositions, Fibonacci Numbers, and Resistor Networks, SIAM Rev., 17 (1975), 465-474.

Index entries for linear recurrences with constant coefficients, signature (3,-4).

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

Example

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

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 *)

Prog

(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 */
(Sage) [lucas_number1(n, 3, 4) for n in range(1, 34)] # Zerinvary Lajos, Apr 23 2009
(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
(PARI) x='x+O('x^100); Vec(1/(1-3*x+4*x^2)) \\ Altug Alkan, Dec 04 2015

Crossrefs

Cf. A006516, A025170, A247563.

Sequence in context: A328878 A178095 A177904 * A059887 A023585 A089948

Adjacent sequences: A049069 A049070 A049071 * A049073 A049074 A049075

Keyword

sign,easy

Author

N. J. A. Sloane

Status

approved