A106853 Expansion of 1/(1 - x + 4*x^2).

1, 1, -3, -7, 5, 33, 13, -119, -171, 305, 989, -231, -4187, -3263, 13485, 26537, -27403, -133551, -23939, 510265, 606021, -1435039, -3859123, 1881033, 17317525, 9793393, -59476707, -98650279, 139256549, 533857665, -23168531, -2158599191, -2065925067

Offset: 0

Author

Paul Barry, May 08 2005

Keywords

• easy
• sign

Comments

Row sums of Riordan array (1,x(1-4x)). In general, a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,n-k)*r^(n-k) yields the row sums of the Riordan array (1,x(1-kx)).

For n >= 1, a(n) equals the determinant of the n X n matrix with 2's along the superdiagonal and the subdiagonal, and 1's along the main diagonal, and 0's everywhere else. - John M. Campbell, Jun 04 2011

For n >= 1, |a(n-1)| is the unique odd positive solution x to 4^(n+1) = 15*x^2 + y^2. The value of y is |A272931(n)|. - Jianing Song, Jan 22 2019

Define the sequence u(n) = (u(n-1) + u(n-2))/u(n-3) with u(1) = 1, u(2) = -1, u(3) = 2. Then u(4*n) = 2*(a(n-1)+4*a(n-2))*a(n-1)/(a(n)+a(n-1))/a(n), u(4*n+1) = a(n+1)/a(n), u(4*n+2) = -1, u(4*n+3) = 4*(a(n)+a(n-1))/(a(n)+a(n+1)). For example, a(2) = -3, a(3) = -7 and u(8) = 5/3, u(9) = 7/3, u(10) = -1. - Michael Somos, Oct 24 2023

Examples

			G.f. = 1 + x - 3*x^2 - 7*x^3 + 5*x^4 + 33*x^5 + 13*x^6 - 119*x^7 - 171*x^8 + ... - _Michael Somos_, Oct 24 2023
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
• Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae (2023). See p. 15.
• Index entries for linear recurrences with constant coefficients, signature (1,-4).

Crossrefs

Cf. A106852, A272931.

Programs

Magma

I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 4*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018

Maple

f:= gfun:-rectoproc({a(n)=a(n-1)-4*a(n-2), a(0)=1,a(1)=1},a(n),remember):
map(f, [$0..100]); # Robert Israel, Jan 15 2018

Mathematica

Join[{a=1,b=1},Table[c=b-4*a;a=b;b=c,{n,80}]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2011 *)
CoefficientList[Series[1/(1-x*(1-4x)),{x,0,40}],x] (* or *) LinearRecurrence[ {1,-4},{1,1},40] (* Harvey P. Dale, May 26 2013 *)
a[ n_] := 2^n * ChebyshevU[n, 1/4]; (* Michael Somos, Oct 24 2023 *)

PARI

x='x+O('x^30); Vec(1/(1-x+4*x^2)) \\ G. C. Greubel, Jan 14 2018

PARI

{a(n) = 2^n*polchebyshev(n, 2, 1/4)}; /* Michael Somos, Oct 24 2023 */

Sage

[lucas_number1(n,1,4) for n in range(1, 36)] # Zerinvary Lajos, Apr 22 2009

Formula

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

a(n) = 2^n*(cos(2*n*arctan(sqrt(15)/5))+sqrt(15)*sin(2*n*arctan(sqrt(15)/5))/15).

a(n) = ((1 + sqrt(-15))^(n+1) - (1 - sqrt(-15))^(n+1))/(2^(n+1)*sqrt(-15)).

a(n) = Sum_{k=0..n} ((-1)^(n-k)*binomial(k, n-k)*4^(n-k)).

a(n) = a(n-1) - 4*a(n-2), a(0) = 1, a(1) = 1. - Philippe Deléham, Oct 21 2008

a(n) = Sum_{k=0..n} A109466(n,k)*4^(n-k). - Philippe Deléham, Oct 25 2008

G.f.: 1/(1 - 2*x)^2/(1 + 3*x*G(0)/2), where G(k) = 1 + 1/(1 - x/(x + (k + 1)/(2*k + 4)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013

For n >= 1, 15*A272931(n)^2 + a(n-1)^2 = 4^(n+1). - Jianing Song, Jan 22 2019

a(n) = Product_{k=1..n} (1 + 4*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019

a(n) = 2^n * U(n, 1/4), where U(n, x) is the Chebyshev polynomial of the second kind. - Federico Provvedi, Mar 28 2022