A007909 Expansion of (1-x)/(1-2*x+x^2-2*x^3).

1, 1, 1, 3, 7, 13, 25, 51, 103, 205, 409, 819, 1639, 3277, 6553, 13107, 26215, 52429, 104857, 209715, 419431, 838861, 1677721, 3355443, 6710887, 13421773, 26843545, 53687091, 107374183, 214748365, 429496729, 858993459, 1717986919, 3435973837, 6871947673

Offset

0,4

Comments

Equals INVERT transform of (1, 0, 2, 2, 2, ...). - Gary W. Adamson, Apr 28 2009

References

Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.

M. E. Larsen, Summa Summarum, A. K. Peters, Wellesley, MA, 2007; see p. 38.

Links

M. F. Hasler, Table of n, a(n) for n = 0..1000 (in replacement of a(0..999) indexed 1..1000 from Vincenzo Librandi).

Charles K. Cook and Michael R. Bacon, Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae, 41 (2013) pp. 27-39.

Shanzhen Gao, Keh-Hsun Chen, Tackling Sequences From Prudent Self-Avoiding Walks, FCS'14, The 2014 International Conference on Foundations of Computer Science.

I. Gessel, Problem 10424, Amer. Math. Monthly, 102 (1995), 70.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 444

Yüksel Soykan, Properties of Generalized (r, s, t, u)-Numbers, Earthline J. of Math. Sci. (2021) Vol. 5, No. 2, 297-327.

Index entries for linear recurrences with constant coefficients, signature (2,-1,2).

Formula

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

a(n) = (1/5)*(2^(n+1)+3*cos(n*Pi/2)+sin(n*Pi/2)).

a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-k-1, 2*k)*2^k. - Paul Barry, Sep 16 2004

a(n) = (1/5)*(2^(n+1) + (-1)^[(n+1)/2] + 2*(-1)^[n/2]). - Ralf Stephan, Jun 09 2005

a(n) = 2*a(n-1)-a(n-2)+2*a(n-3). Sequence is identical to its half second differences from the second term; a(n)+a(n+2)=2^(n+1). - Paul Curtz, Dec 17 2007

a(n+1) = (2^n)*Sum_{k=1..n} (-1)^floor(k/2)/2^k. - Yalcin Aktar, Jul 20 2008

Maple

U:=n->(1/5)*(2^(n+1)+3*cos(n*Pi/2)+sin(n*Pi/2)); [seq(U(n), n=0..50)];

Mathematica

CoefficientList[Series[(1-x)/(1-2*x+x^2-2*x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 17 2012 *)
LinearRecurrence[{2, -1, 2}, {1, 1, 1}, 40] (* Harvey P. Dale, Jul 26 2016 *)

Prog

(Magma) I:=[1, 1, 1]; [n le 3 select I[n] else 2*Self(n-1)-Self(n-2)+2*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jun 17 2012
(PARI) a(n)=2^(n+1)\5+(n%4<2) \\ M. F. Hasler, Feb 22 2018

Crossrefs

Cf. A005251, A007679, A007910.

Sequence in context: A169914 A078000 A190569 * A363143 A282913 A284026

Adjacent sequences: A007906 A007907 A007908 * A007910 A007911 A007912

Keyword

nonn,easy

Author

Mogens Esrom Larsen (mel(AT)math.ku.dk)

Extensions

Offset corrected by M. F. Hasler, Feb 22 2018

Status

approved