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A027934
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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).
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21
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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
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OFFSET
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0,3
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COMMENTS
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Number of compositions of n with at least one even part (offset 2). - Vladeta Jovovic, Dec 29 2004
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) 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
a(n-1) is the number of subsets of {1,2,..,n} that contain n that have at least one pair of consecutive integers. For example, for n=5, a(4) = 11 and the 11 subsets are {4,5}, {1,2,5}, {1,4,5}, {2,3,5}, {2,4,5}, {3,4,5}, {1,2,3,5}, {1,2,4,5}, {1,3,4,5}, {2,3,4,5}, {1,2,3,4,5}. Note that A008466(n) is the number of all subsets of {1,2,..,n} that have at least one pair of consecutive integers. - Enrique Navarrete, Aug 15 2020
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LINKS
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FORMULA
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a(n) = Sum_{j=0..floor(n/2)} Sum_{k=0..n-2*j} binomial(n-j, n-2*j-k). - Paul Barry, Feb 07 2003
G.f.: x*(1-x)/((1-2*x)*(1-x-x^2)).
a(n) = 2^n - Fibonacci(n+1). (End) - 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) = 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) = 4*a(n-2) + A000032(n-2). (End)
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MAPLE
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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
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MATHEMATICA
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nn=31; a:=1/(1-x-x^2); b:=1/(1-2x); CoefficientList[Series[a*x*(1+x*b), {x, 0, nn}], x] (* Geoffrey Critzer, Jan 04 2014 *)
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PROG
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(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
(Magma) [2^n - Fibonacci(n+1): n in [0..35]]; // G. C. Greubel, Sep 27 2019
(Sage) [2^n - fibonacci(n+1) for n in (0..35)] # G. C. Greubel, Sep 27 2019
(GAP) List([0..35], n-> 2^n - Fibonacci(n+1) ); # G. C. Greubel, Sep 27 2019
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CROSSREFS
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T(n, n) + T(n, n+1) + ... + T(n, 2n), T given by A027926.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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