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A081186 4th binomial transform of (1,0,1,0,1,......), A059841. 11
1, 4, 17, 76, 353, 1684, 8177, 40156, 198593, 986404, 4912337, 24502636, 122336033, 611148724, 3054149297, 15265963516, 76315468673, 381534296644, 1907542343057, 9537324294796, 47685459212513, 238423809278164, 1192108586037617, 5960511549128476 (list; graph; refs; listen; history; text; internal format)
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/2 + 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

MATHEMATICA

CoefficientList[Series[(1 - 4 x)/((1 - 3 x) (1 - 5 x)), {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 *)

CROSSREFS

Cf. A005059, A007582.

Sequence in context: A117439 A081910 A026773 * A239204 A005572 A202879

Adjacent sequences:  A081183 A081184 A081185 * A081187 A081188 A081189

KEYWORD

nonn,easy

AUTHOR

Paul Barry, Mar 11 2003

STATUS

approved

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Last modified October 21 08:47 EDT 2019. Contains 328292 sequences. (Running on oeis4.)