OFFSET
0,2
COMMENTS
Binomial transform of A081187.
a(n) is also the number of words of length n over an alphabet of six letters, of which a chosen one appears an even number of times. See a comment in A007582, also for the crossrefs, for the 1- to 11-letter word cases. - Wolfdieter Lang, Jul 17 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)=1, a(1)=5.
G.f.: (1-5*x)/((1-4*x)*(1-6*x)).
E.g.f.: exp(5*x)*cosh(x).
a(n) = (4^n + 6^n)/2.
a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*5^(n-2k).
E.g.f.: exp(5*x)*cosh(x) = (1/2)*E(0), where E(k) = 1 + 2^k/(3^k - 6*x*9^k/(6*x*3^k + (k+1)*2^k/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011
a(n) = A074612(n)/2. - G. C. Greubel, Jan 13 2024
MAPLE
seq( (4^n + 6^n)/2, n=0..25); # G. C. Greubel, Dec 26 2019
MATHEMATICA
CoefficientList[Series[(1-5x)/((1-4x)(1-6x)), {x, 0, 25}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{10, -24}, {1, 5}, 26] (* G. C. Greubel, Dec 26 2019 *)
PROG
(Magma) [4^n/2 + 6^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013
(PARI) vector(26, n, (4^(n-1) + 6^(n-1))/2) \\ G. C. Greubel, Dec 26 2019
(Sage) [(4^n + 6^n)/2 for n in (0..25)] # G. C. Greubel, Dec 26 2019
(GAP) List([0..25], n-> (4^n + 6^n)/2); # G. C. Greubel, Dec 26 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Mar 11 2003
STATUS
approved