A006904 a(n) = a(n-1) + 2*a(n-2) + (-1)^n. (Formerly M3254)

1, 1, 4, 5, 14, 23, 52, 97, 202, 395, 800, 1589, 3190, 6367, 12748, 25481, 50978, 101939, 203896, 407773, 815566, 1631111, 3262244, 6524465, 13048954, 26097883, 52195792, 104391557, 208783142, 417566255, 835132540, 1670265049, 3340530130, 6681060227

Offset

4,3

References

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 327.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2d edition 1994, p. 341.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=4..37.

Index entries for linear recurrences with constant coefficients, signature (0,3,2).

Formula

G.f.: x^4*(-x^2 - x - 1)/((1 + x)^2*(2*x - 1)). - corrected by Harvey P. Dale, Apr 22 2011

With offset 0: a(n) = 1/9*(7*2^n+(-1)^n*(3*n+2)); if b(1)=1, b(k) = 2*b(k-1)+(-1)^k*k, then for n>4, a(n)=b(n-4). - Benoit Cloitre, Oct 28 2002

a(n) = 3*a(n-2)+2*a(n-3). - Harvey P. Dale, Apr 22 2011

Mathematica

LinearRecurrence[{0, 3, 2}, {1, 1, 4}, 41] (* or *) CoefficientList[Series[ (-x^2-x-1)/((1+x)^2 (2x-1)), {x, 0, 40}], x] (* Harvey P. Dale, Apr 22 2011 *)

Crossrefs

Sequence in context: A246984 A306897 A348889 * A200177 A241863 A350527

Adjacent sequences: A006901 A006902 A006903 * A006905 A006906 A006907

Keyword

nonn,easy

Author

Simon Plouffe and N. J. A. Sloane

Extensions

More terms from Jason Yuen, Aug 25 2025

Status

approved