OFFSET
0,3
COMMENTS
Binomial transform of A005059.
Conjecture (verified up to a(9)): Number of collinear 4-tuples of points in a 4 X 4 X 4 X ... n-dimensional cubic grid. - R. H. Hardin, May 24 2010
a(n) is also the total number of words of length n, over an alphabet of six letters, of which one of them appears an odd number of times. See a Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter case), and the Balakrishnan reference there. For the 2-, 3-, 5- and 7-letter case analogs see A131577, A003462, A005059 and A081200, respectively. - Wolfdieter Lang, Jul 16 2017
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (10,-24).
FORMULA
a(n) = 10*a(n-1) - 24*a(n-2) with n>1, a(0)=0, a(1)=1.
G.f.: x/((1-4*x)*(1-6*x)).
a(n) = 6^n/2 - 4^n/2.
E.g.f.: exp(4*x)*(exp(2*x) - 1)/2. - Stefano Spezia, Jul 23 2024
MAPLE
seq(add(2^(2*n-k)*binomial(n, k)/2, k=1..n), n=0..20); # Zerinvary Lajos, Apr 18 2009
MATHEMATICA
CoefficientList[Series[x / ((1 - 4 x) (1 - 6 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{10, -24}, {0, 1}, 21] (* Michael De Vlieger, Jul 16 2017 *)
PROG
(Magma) [6^n/2-4^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Mar 11 2003
STATUS
approved