A087168 Expansion of (1 + 2*x)/(1 + 3*x + 4*x^2).

1, -1, -1, 7, -17, 23, -1, -89, 271, -457, 287, 967, -4049, 8279, -8641, -7193, 56143, -139657, 194399, -24569, -703889, 2209943, -3814273, 2603047, 7447951, -32756041, 68476319, -74404793, -50690897, 449691863, -1146312001, 1640168551, -335257649, -5554901257

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Aug 22 2003

Keywords

• easy
• sign

Comments

For positive n, a(n) equals 2^n times the permanent of the (2n) X (2n) tridiagonal matrix with 1/sqrt(2)'s along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit). - John M. Campbell, Jul 08 2011

For n > 3, equals -1 times the determinant of the (n-2) X (n-2) matrix with 2^2's along the superdiagonal, 3^2's along the main diagonal, 4^2's along the subdiagonal, etc., and 0's everywhere else. - John M. Campbell, Dec 01 2011

Examples

			G.f. = 1 - x - x^2 + 7*x^3 - 17*x^4 + 23*x^5 - x^6 - 89*x^7 + 271*x^8 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 14.
• Index entries for linear recurrences with constant coefficients, signature (-3,-4).

Crossrefs

Cf. A001607, A049072.

Programs

Magma

A087168:= func< n | &+[ Binomial(n+k, 2*k)*(-2)^(n-k): k in [0..n] ] >;
[A087168(n): n in [0..35]];

Mathematica

CoefficientList[Series[(1+2x)/(1+3x+4x^2), {x, 0, 30}], x]
Table[-Det[Array[Sum[KroneckerDelta[#1, #2+q]*(q+3)^2, {q, -1, n-2}] &, {n-2, n-2}]], {n, 4, 50}] (* John M. Campbell, Dec 01 2011 *)
LinearRecurrence[{-3,-4},{1,-1},40] (* Harvey P. Dale, Apr 23 2014 *)

PARI

{a(n) = real( (-1 - quadgen(-7))^n )}; /* Michael Somos, Sep 19 2014 */

SageMath

def A087168(n): return (-2)^(n-1)*(2*chebyshev_U(n-2, 3/4) -chebyshev_U(n-1, 3/4))
[A087168(n) for n in (0..50)] # G. C. Greubel, Jun 09 2022

Formula

G.f.: (1+2*x)/(1+3*x+4*x^2).

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

a(n) = Sum_{k=0..n} C(n+k,2*k)*(-2)^(n-k).

a(n) = -a(-1-n) * 2^(2*n+1) = A001607(2*n + 1) for all n in Z. - Michael Somos, Sep 19 2014

a(n) = (-2)^(n-1)*(2*ChebyshevU(n-2, 3/4) - ChebyshevU(n-1, 3/4)). - G. C. Greubel, Jun 09 2022