

A077949


Expansion of 1/(1x2*x^3).


15



1, 1, 1, 3, 5, 7, 13, 23, 37, 63, 109, 183, 309, 527, 893, 1511, 2565, 4351, 7373, 12503, 21205, 35951, 60957, 103367, 175269, 297183, 503917, 854455, 1448821, 2456655, 4165565, 7063207, 11976517, 20307647, 34434061, 58387095, 99002389, 167870511, 284644701
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OFFSET

0,4


COMMENTS

Row sums of the Riordan array (1, x*(1+2*x^2)).  Paul Barry, Jan 12 2006
The compositions of n in which each natural number is colored by one of p different colors are called pcolored compositions of n. For n>=3, 3*a(n3) equals the number of 3colored compositions of n with all parts >=3, such that no adjacent parts have the same color.  Milan Janjic, Nov 27 2011
Number of compositions of n into parts 1 and two sorts of parts 2.  Joerg Arndt, Aug 29 2013
a(n+2) equals the number of words of length n on alphabet {0,1,2}, having at least two zeros between every two successive nonzero letters.  Milan Janjic, Feb 07 2015
Number of pairs of rabbits when there are 2 pairs per litter and offspring reach parenthood after 3 gestation periods; a(n) = a(n1) + 2*a(n3), with a(0) = a(1) = a(2) = 1.  Robert FERREOL, Oct 27 2018


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,2).


FORMULA

a(n) = Sum_{k=0..floor(n/2)} C(n2k, k)*2^k.  Paul Barry, Nov 18 2003
a(n) = Sum_{k=0..n} C(k, floor((nk)/2))*2^((nk)/2)*(1+(1)^(nk))/2.  Paul Barry, Jan 12 2006
a(n) = term (1,1) in the 3x3 matrix [1,1,0; 0,0,1; 2,0,0]^n.  Alois P. Heinz, Aug 16 2008
G.f.: Q(0)/2, where Q(k) = 1 + 1/(1  x*(2*k+1 + 2*x^2)/( x*(2*k+2 + 2*x^2) + 1/Q(k+1) )); (continued fraction).  Sergei N. Gladkovskii, Aug 29 2013


MAPLE

a:= n> (<<110>, <001>, <200>>^n)[1, 1]:
seq(a(n), n=0..40); # Alois P. Heinz, Aug 16 2008


MATHEMATICA

CoefficientList[Series[1/(1x2*x^3), {x, 0, 50}], x] (* JeanFrançois Alcover, Mar 11 2014 *)
LinearRecurrence[{1, 0, 2}, {1, 1, 1}, 50] (* Robert G. Wilson v, Jul 12 2014 *)


PROG

(PARI) Vec(1/(1x2*x^3)+O(x^50)) \\ Charles R Greathouse IV, Sep 23 2012
(Magma) [n le 3 select 1 else Self(n1)+2*Self(n3): n in [1..50]]; // Vincenzo Librandi, Mar 13 2014
(Sage) (1/(1x2*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jun 22 2019
(GAP) a:=[1, 1, 1];; for n in [4..30] do a[n]:=a[n1]+2*a[n3]; od; a; # G. C. Greubel, Jun 22 2019


CROSSREFS

Unsigned version of A077974. Cf. A003229.
Sequence in context: A125272 A127443 A003229 * A077974 A126273 A007658
Adjacent sequences: A077946 A077947 A077948 * A077950 A077951 A077952


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Nov 17 2002


STATUS

approved



