A007613 a(n) = (8^n + 2*(-1)^n)/3. (Formerly M2129)

1, 2, 22, 170, 1366, 10922, 87382, 699050, 5592406, 44739242, 357913942, 2863311530, 22906492246, 183251937962, 1466015503702, 11728124029610, 93824992236886, 750599937895082, 6004799503160662, 48038396025285290, 384307168202282326, 3074457345618258602

Offset

0,2

References

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

D. S. Clark, Proof without words, Math. Mag., 63 (1990), 29.

Index entries for linear recurrences with constant coefficients, signature (7,8).

Formula

a(n) = A078008(3*n). - Paul Barry, Nov 29 2003

From Paul Barry, Mar 24 2004: (Start)

a(n) = (A082311(n) + (-1)^n)/2.

a(n) = (A001045(3*n+1) + (-1)^n)/2. (End)

a(n) = Sum_{k=0..n} binomial(3*n, 3*k). - Paul Barry, Jan 13 2005

a(n) = 8*a(n-1) + 6*(-1)^n. - Paul Curtz, Nov 19 2007

From Colin Barker, Sep 29 2014: (Start)

a(n) = 7*a(n-1) + 8*a(n-2).

G.f.: (1-5*x) / ((1+x)*(1-8*x)). (End)

E.g.f.: (1/3)*(exp(8*x) + 2*exp(-x)). - G. C. Greubel, Apr 23 2023

Mathematica

LinearRecurrence[{7, 8}, {1, 2}, 41] (* G. C. Greubel, Apr 23 2023 *)

Prog

(PARI) a(n)=(8^n + 2*(-1)^n)/3 \\ Charles R Greathouse IV, Jun 06, 2011
(Magma) [(8^n + 2*(-1)^n)/3: n in [0..30]]; // Vincenzo Librandi, Aug 14 2011
(PARI) Vec((5*x-1)/((x+1)*(8*x-1)) + O(x^50)) \\ Colin Barker, Sep 29 2014
(SageMath) [(8^n -4*(n%2) +2)/3 for n in range(41)] # G. C. Greubel, Apr 23 2023

Crossrefs

Cf. A001045, A006566, A078008, A082311, A139459.

Sequence in context: A230835 A270407 A000184 * A346796 A279801 A043037

Adjacent sequences: A007610 A007611 A007612 * A007614 A007615 A007616

Keyword

nonn,easy

Author

N. J. A. Sloane, Robert G. Wilson v

Extensions

More terms from Colin Barker, Sep 29 2014

Status

approved