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A078012
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Expansion of (1 - x) / (1 - x - x^3) in powers of x.
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12
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1, 0, 0, 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023, 5896, 8641, 12664, 18560, 27201, 39865, 58425, 85626, 125491, 183916, 269542, 395033, 578949, 848491, 1243524, 1822473, 2670964, 3914488, 5736961, 8407925
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OFFSET
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0,7
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COMMENTS
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Number of compositions of n into parts >=3. [From Milan Janjic, Jun 28 2010]
From Adi Dani, May 22 2011: (Start)
Number of compositions of number n into parts of the form 3*k+1, k>=0.
For example, a(10)=19 and all compositions of 10 in parts 1,4,7 or 10 are
(1,1,1,1,1,1,1,1,1,1), (1,1,1,1,1,1,4), (1,1,1,1,1,4,1), (1,1,1,1,4,1,1), (1,1,1,4,1,1,1), (1,1,4,1,1,1,1), (1,4,1,1,1,1,1), (4,1,1,1,1,1,1), (1,1,4,4), (1,4,1,4), (1,4,4,1), (4,1,1,4),(4,1,4,1), (4,4,1,1), (1,1,1,7), (1,1,7,1), (1,7,1,1), (7,1,1,1), (10). (End)
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REFERENCES
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C. K. Fan, Structure of a Hecke algebra quotient. J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. [Page 156, f_n.]
Taylor L. Booth, Sequential Machines and Automata Theory, John Wiley and Sons, Inc., 1967, page 331ff.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index to sequences with linear recurrences with constant coefficients, signature (1,0,1).
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FORMULA
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a(n) = sum(binomial(n-3-2*i, i), i=0..(n-3)/3), n>=1, a(0) = 1.
From Michael Somos, May 03 2011: (Start)
Euler transform of A065417.
G.f.: (1 - x) / (1 - x - x^3). a(n) = a(n-1) + a(n-3).
a(-n) = A077961(n). a(n+3) = A000930(n). a(n+5) = A068921(n). (End)
a(n+1) = A013979(n-3) + A135851(n) + A107458(n), n>=3.
a(n) = a(n-1) + a(n-3) for n >= 4. - Jaroslav Krizek, May 07 2011.
G.f. 1/(1-sum(k>=3, x^k)). [Joerg Arndt, Aug 13 2012]
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EXAMPLE
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1 + x^3 + x^4 + x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 6*x^9 + 9*x^10 + 13*x^11 + ...
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MAPLE
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A078012 := proc(n): if n=0 then 1 else add(binomial(n-3-2*i, i), i=0..(n-3)/3) fi: end: seq(A078012(n), n=0..46); # [Johannes W. Meijer, Aug 11 2011]
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MATHEMATICA
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CoefficientList[ Series[(1 - x)/(1 - x - x^3), {x, 0, 46}], x] (* Robert G. Wilson v, May 25 2011 *)
LinearRecurrence[{1, 0, 1}, {1, 0, 0}, 70] (* From Vladimir Joseph Stephan Orlovsky, Feb 24 2012 *)
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PROG
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(PARI) {a(n) = if( n<0, n = -n; polcoeff( 1 / (1 + x^2 - x^3) + x * O(x^n), n) , polcoeff( (1 - x) / (1 - x - x^3) + x * O(x^n), n))} /* Michael Somos May 03 2011 */
(Haskell)
a078012 n = a078012_list !! n
a078012_list = 1 : 0 : 0 : 1 : zipWith (+) a078012_list
(zipWith (+) (tail a078012_list) (drop 2 a078012_list))
-- Reinhard Zumkeller, Mar 23 2012
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CROSSREFS
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Cf. A000930, A065417, A068921, A077961.
Cf. A135851.
Sequence in context: A068921 A000930 * A135851 A199804 A101913 A121653
Adjacent sequences: A078009 A078010 A078011 * A078013 A078014 A078015
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane, Nov 17 2002, Mar 08 2008
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STATUS
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approved
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