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A081187 5th binomial transform of (1,0,1,0,1,...), A059841. 5
1, 5, 26, 140, 776, 4400, 25376, 148160, 872576, 5169920, 30757376, 183495680, 1096779776, 6563901440, 39316299776, 235629363200, 1412702437376, 8471919656960, 50814338072576, 304817308958720, 1828628975845376 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A081569 A005573 A081911 * A182401 A363308 A104498
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Mar 11 2003
STATUS
approved

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Last modified March 18 22:47 EDT 2024. Contains 370951 sequences. (Running on oeis4.)