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A074331 a(n) = Fibonacci(n+1) - (1 + (-1)^n)/2. 9
0, 1, 1, 3, 4, 8, 12, 21, 33, 55, 88, 144, 232, 377, 609, 987, 1596, 2584, 4180, 6765, 10945, 17711, 28656, 46368, 75024, 121393, 196417, 317811, 514228, 832040, 1346268, 2178309, 3524577, 5702887, 9227464, 14930352, 24157816, 39088169 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

a(n) is the convolution of F(n) with the sequence (1,0,1,0,1,0,...).

Transform of F(n) under the Riordan array (1/(1-x^2), x). - Paul Barry, Apr 16 2005

LINKS

Table of n, a(n) for n=0..37.

Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-1)

FORMULA

a(n) = Sum_{i=0..floor(n/2)} Fibonacci(2i + e), where e = 2*(n/2 - floor(n/2)).

a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n > 3, a(0)=0, a(1)=1, a(2)=1, a(3)=3.

G.f.: x / ( (x-1)*(1+x)*(x^2+x-1) ).

a(2n+1) = Fibonacci(2n+2), a(2n) = Fibonacci(2n+1)-1.

a(n-1) = Sum_{k=0..floor(n/2)} binomial(n-k, k-1). - Paul Barry, Jul 07 2004

a(n) = Sum_{k=0..floor((n-1)/2)} Fibonacci(n-2k). - Paul Barry, Apr 16 2005

a(n) = Sum_{k=0..n} Fibonacci(k)*(1-(-1)^floor((n+k-1)/2)). - Paul Barry, Apr 16 2005

a(n) = Fibonacci(n) + a(n-2) for n > 1. - Zerinvary Lajos, Mar 17 2008

a(n) = floor(g^(n+1)/sqrt(5)), where g = (sqrt(5) + 1)/2. - Federico Provvedi, Mar 27 2013

MAPLE

with(combinat):seq(fibonacci(n+1)-(1+(-1)^n)/2, n=0..40); # Zerinvary Lajos, Mar 17 2008

MATHEMATICA

CoefficientList[Series[x/(1-x-2*x^2+x^3+x^4), {x, 0, 40}], x]

Table[Floor[GoldenRatio^(k+1)/Sqrt[5]], {k, 0, 40}] (* Federico Provvedi, Mar 26 2013 *)

PROG

(PARI) a(n)=if(n<0, 0, fibonacci(n+1)-(n+1)%2)

CROSSREFS

Essentially the same as A052952.

Sequence in context: A145722 A147622 A173534 * A052952 A245121 A153339

Adjacent sequences:  A074328 A074329 A074330 * A074332 A074333 A074334

KEYWORD

nonn,easy

AUTHOR

Mario Catalani (mario.catalani(AT)unito.it), Aug 21 2002

STATUS

approved

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Last modified April 26 11:46 EDT 2019. Contains 322472 sequences. (Running on oeis4.)