OFFSET
0,2
COMMENTS
Binomial transform of A007582.
a(n) is a companion to A005059(n): a(n) + A005059(n) = 5^n; e.g. a(4) = A005059(4) = 353 + 272 = 625 = 5^4. - Gary W. Adamson, Jun 30 2006
Number of words of length n from an alphabet of 5 letters in which a chosen letter appears an even number of times. - James Mahoney, Feb 03 2012 [See a comment in A007582, also for crossrefs. for the 1- to 11-letter word cases. - Wolfdieter Lang, Jul 17 2017]
The sequence of fractions x(n) = a(n+1)/a(n) satisfies a simple recurrence x(n+1) = 108 - (815 - 1500 / x(n-1)) / x(n) known as Muller's recurrence. It is used for the demonstration of an unexpected failure of floating-point computations. - Andrey Zabolotskiy, Sep 17 2019
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Christian Hill, Muller's Recurrence, 2017.
Index entries for linear recurrences with constant coefficients, signature (8,-15).
FORMULA
a(n) = 8*a(n-1) - 15*a(n-2) with n>1, a(0)=1, a(1)=4.
G.f.: (1-4*x)/((1-3*x)*(1-5*x)).
a(n) = (3^n + 5^n)/2.
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*4^(n-2*k).
E.g.f.: exp(4*x) * cosh(x). - Paul Barry, Oct 06 2004
EXAMPLE
Say the alphabet is {a,b,c,d,e} and we want to know how many words of length one and two contain c an even number of times. a(1) = 4, which we can see by the four words {(a),(b),(d),(e)} and a(2) = 17, which we can see by the seventeen words {(a,a), (a,b), (a,d), (a,e), (b,a), (b,b), (b,d), (b,e), (c,c), (d,a), (d,b), (d,d), (d,e), (e,a), (e,b), (e,d), (e,e)}. - James Mahoney, Feb 03 2012
MAPLE
seq( (3^n + 5^n)/2, n=0..30); # G. C. Greubel, Dec 26 2019
MATHEMATICA
CoefficientList[Series[(1-4x)/((1-3x)(1-5x)), {x, 0, 25}], x] (* Vincenzo Librandi, Aug 07 2013 *)
LinearRecurrence[{8, -15}, {1, 4}, 30] (* Harvey P. Dale, Apr 13 2019 *)
PROG
(Magma) [3^n/2+5^n/2: n in [0..25]]; // Vincenzo Librandi, Aug 07 2013
(PARI) vector(31, n, (3^(n-1) + 5^(n-1))/2 ) \\ G. C. Greubel, Dec 26 2019
(Sage) [(3^n + 5^n)/2 for n in (0..25)] # G. C. Greubel, Dec 26 2019
(GAP) List([0..25], n-> (3^n + 5^n)/2); # G. C. Greubel, Dec 26 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Mar 11 2003
STATUS
approved