A133631 a(n) = a(n-1) - 4*a(n-2), a(0)=1, a(1)=2.

1, 2, -2, -10, -2, 38, 46, -106, -290, 134, 1294, 758, -4418, -7450, 10222, 40022, -866, -160954, -157490, 486326, 1116286, -829018, -5294162, -1978090, 19198558, 27110918, -49683314, -158126986, 40606270, 673114214, 510689134, -2181767722, -4224524258

Offset

0,2

Links

Table of n, a(n) for n=0..32.

Index entries for linear recurrences with constant coefficients, signature (1,-4).

Formula

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

a(n) = Sum_{k=0..n} A133607(n,k)*2^k. - Philippe Deléham, Dec 29 2007

a(n) = 2^n*U(n, 1/4) + 2^(n-1)*U(n-1, 1/4) = A106853(n) + A106853(n-1) where U is the Chebyshev polynomial of the 2nd kind. - Michael Somos, Oct 24 2023

Example

G.f. = 1 + 2*x - 2*x^2 - 10*x^3 - 2*x^4 + 38*x^5 + 46*x^6 - 106*x^7 + ... - Michael Somos, Oct 24 2023

Mathematica

a[ n_] := 2^n * ChebyshevU[n, 1/4] + 2^(n-1) * ChebyshevU[n-1, 1/4]; (* Michael Somos, Oct 24 2023 *)

Prog

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

Crossrefs

Cf. A106853, A133607.

Sequence in context: A297793 A351177 A319880 * A137450 A344998 A321415

Adjacent sequences: A133628 A133629 A133630 * A133632 A133633 A133634

Keyword

easy,sign

Author

Philippe Deléham, Dec 28 2007

Status

approved