OFFSET
0,2
COMMENTS
Also, the cogrowth sequence of C3 X C3 = <S,T | S^3, T^3, STS^2T^2>; 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).
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Colin Barker, Sep 29 2014
STATUS
approved