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

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

Offset: 0

Author

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

Keywords

• nonn
• easy

Comments

Also, the cogrowth sequence of C3 X C3 = ; that is, the number of words of length 3n that reduce to the identity. - Sean A. Irvine, Nov 04 2024

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).

Crossrefs

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

Programs

Magma

[(8^n + 2*(-1)^n)/3: n in [0..30]]; // Vincenzo Librandi, Aug 14 2011

Mathematica

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

PARI

a(n)=(8^n + 2*(-1)^n)/3 \\ Charles R Greathouse IV, Jun 06 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

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

Extensions

More terms from Colin Barker, Sep 29 2014