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A214727 a(n) = a(n-1) + a(n-2) + a(n-3) with a(0)=1, a(1) = a(2) = 2. 55
1, 2, 2, 5, 9, 16, 30, 55, 101, 186, 342, 629, 1157, 2128, 3914, 7199, 13241, 24354, 44794, 82389, 151537, 278720, 512646, 942903, 1734269, 3189818, 5866990, 10791077, 19847885, 36505952, 67144914, 123498751, 227149617, 417793282 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Part of a group of sequences defined by a(0), a(1)=a(2), a(n) = a(n-1) + a(n-2) + a(n-3) which is a sub-group of sequences with linear recurrences and constant coefficients listed in the index.

Note: A000073 (with offset=1), 1 followed by A000073, A000213, A141523, A214727, A214825 to A214831 completely define possible sequences with a(0)=0,1,2...9 and a(1)=a(2)=0,1,2...9 excluding any multiples of these sequences and the trivial case of a(0)=a(1)=a(2)=0.

Note: allowing a(0)=0 and a(1)=a(2)=1,2,3....9 leads to A000073 (with offset=1) and its multiples.

Note: allowing a(0)=1,2,3....9 a(1)=a(2)=0 leads to 1 followed by A000073 and its multiples.

With offset of 6 this sequence is the 8th row of tribonacci array A136175.

LINKS

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

Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

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

FORMULA

G.f.: (1+x-x^2)/(1-x-x^2-x^3).

a(n) = K(n) -2*T(n+1) + 3*T(n), where K(n) = A001644(n), T(n) = A000073(n+1). - G. C. Greubel, Apr 23 2019

EXAMPLE

G.f. = 1 + 2*x + 2*x^2 + 5 x^3 + 9*x^4 + 16*x^5 + 30*x^6 + 55*x^7 + ...

MATHEMATICA

LinearRecurrence[{1, 1, 1}, {1, 2, 2}, 40] (* Ray Chandler, Dec 08 2013 *)

PROG

(Haskell)

a214727 n = a214727_list !! n

a214727_list = 1 : 2 : 2 : zipWith3 (\x y z -> x + y + z)

   a214727_list (tail a214727_list) (drop 2 a214727_list)

-- Reinhard Zumkeller, Jul 31 2012

(PARI) a(n)=([0, 1, 0; 0, 0, 1; 1, 1, 1]^n*[1; 2; 2])[1, 1] \\ Charles R Greathouse IV, Mar 22 2016

(PARI) my(x='x+O('x^40)); Vec((1+x-x^2)/(1-x-x^2-x^3)) \\ G. C. Greubel, Apr 23 2019

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+x-x^2)/(1-x-x^2-x^3) )); // G. C. Greubel, Apr 23 2019

(Sage) ((1+x-x^2)/(1-x-x^2-x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 23 2019

(GAP) a:=[1, 2, 2];; for n in [4..40] do a[n]:=a[n-1]+a[n-2]+a[n-3]; od; a; # G. C. Greubel, Apr 23 2019

CROSSREFS

Cf. A000213, A000288, A000322, A000383, A060455, A136175, A141036, A141523, A214825-A214831.

Sequence in context: A325104 A054229 A212812 * A302483 A052969 A002990

Adjacent sequences:  A214724 A214725 A214726 * A214728 A214729 A214730

KEYWORD

nonn,easy

AUTHOR

Abel Amene, Jul 27 2012

STATUS

approved

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Last modified January 21 10:59 EST 2020. Contains 331105 sequences. (Running on oeis4.)