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A027934 a(0)=0, a(1)=1, a(2)=2; for n > 2, a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3). 21
0, 1, 2, 5, 11, 24, 51, 107, 222, 457, 935, 1904, 3863, 7815, 15774, 31781, 63939, 128488, 257963, 517523, 1037630, 2079441, 4165647, 8342240, 16702191, 33433039, 66912446, 133899917, 267921227, 536038872, 1072395555, 2145305339 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of compositions of n with at least one even part (offset 2). - Vladeta Jovovic, Dec 29 2004

First differences of A008466. a(n) = A008466(n+2) - A008466(n+1). - Alexander Adamchuk, Apr 06 2006

Starting with "1" = eigensequence of a triangle with the Fibonacci series as the left border and the rest 1's. - Gary W. Adamson, Jul 24 2010

An elephant sequence, see A175654. For the corner squares 24 A[5] vectors, with decimal values between 11 and 416, lead to this sequence (without the leading 0). For the central square these vectors lead to the companion sequence A099036 (without the first leading 1). - Johannes W. Meijer, Aug 15 2010

a(n) = Sum_{k=1..n} A108617(n,k) / 2. - Reinhard Zumkeller, Oct 07 2012

a(n) is the number of binary strings that contain the substring 11 or end in 1. a(3) = 5 because we have: 001, 011, 101, 110, 111. - Geoffrey Critzer, Jan 04 2014

a(n-1), n >= 1, is the number of nonexisting (due to the maturation delay) "[male-female] pairs of Fibonacci rabbits" at the beginning of the n-th month. - Daniel Forgues, May 06 2015

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.

OEIS Wiki, Fibonacci rabbits

Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

FORMULA

a(n) = sum(sum(C(n-j,n-2j-k), k=0..n-2j), j=0..floor(n/2)). - Paul Barry, Feb 07 2003

Row sums of A105809. G.f.: x(1-x)/((1-2x)(1-x-x^2)); a(n) = 2^n - Fibonacci(n+1). - Paul Barry, Jan 23 2004, corrected Apr 06 2006 and Oct 05 2012

a(n) = sum{j=0..n, sum{k=0..n, binomial(n-k, k+j)}}. - Paul Barry, Aug 29 2004

a(n) = (sum of (n+1)-th row of the triangle in A108617) / 2. - Reinhard Zumkeller, Jun 12 2005

a(n) = term (1,1) - term (2,2) in the 3 X 3 matrix [2,0,0; 0,1,1; 0,1,0]^n. - Alois P. Heinz, Jul 28 2008

a(n) = 2^n - (1/2)*(1/2 + (1/2)*sqrt(5))^n - (1/10)*(1/2 +(1/2)*sqrt(5))^n *sqrt(5) + (1/10)*sqrt(5)*(1/2 - (1/2)*sqrt(5))^n - (1/2)*(1/2 - (1/2)*sqrt(5))^n, with n >= 0. - Paolo P. Lava, Oct 02 2008

a(n) = 2^n - A000045(n+1). - Geoffrey Critzer, Jan 04 2014

a(n) ~ 2^n. - Daniel Forgues, May 06 2015

From Bob Selcoe, Mar 29 2016: (Start)

a(n) = 2*a(n-1) + A000045(n-2).

a(n) = 4*a(n-2) + A000032(n-2).

(End)

MAPLE

A027934:= proc(n) local K; K:= Matrix ([[2, 0, 0], [0, 1, 1], [0, 1, 0]])^n; K[1, 1]-K[2, 2] end: seq (A027934(n), n=0..31); # Alois P. Heinz, Jul 28 2008

a := n -> 2^n - combinat:-fibonacci(n+1): seq(a(n), n=0..31); # Peter Luschny, May 09 2015

MATHEMATICA

nn=31; a=1/(1-x-x^2); b=1/(1-2x); CoefficientList[Series[a x+a x^2b, {x, 0, nn}], x] (* Geoffrey Critzer, Jan 04 2014 *)

LinearRecurrence[{3, -1, -2}, {0, 1, 2}, 32] (* Jean-Fran├žois Alcover, Jan 09 2016 *)

PROG

(Haskell)

a027934 n = a027934_list !! n

a027934_list = 0 : 1 : 2 : zipWith3 (\x y z -> 3 * x - y - 2 * z)

               (drop 2 a027934_list) (tail a027934_list) a027934_list

-- Reinhard Zumkeller, Oct 07 2012

(PARI) a(n)=2^n-fibonacci(n+1) \\ Charles R Greathouse IV, Jun 11 2015

CROSSREFS

Row sums of triangle A131767. - Gary W. Adamson, Jul 13 2007

a(n) = A101220(1, 2, n+1).

T(n, n) + T(n, n+1) + ... + T(n, 2n), T given by A027926.

Diagonal sums of A055248.

Cf. A000045, A000079, A008466, A059570, A099036, A047967, A000032.

Sequence in context: A294378 A090764 A182557 * A134389 A286945 A111297

Adjacent sequences:  A027931 A027932 A027933 * A027935 A027936 A027937

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling

EXTENSIONS

Simpler definition from Miklos Kristof, Nov 24 2003

Initial zero added by N. J. A. Sloane, Feb 13 2008

Definition fixed by Reinhard Zumkeller, Oct 07 2012

STATUS

approved

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Last modified October 22 14:36 EDT 2018. Contains 316486 sequences. (Running on oeis4.)