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A077947
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Expansion of 1/(1 - x - x^2 - 2*x^3).
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20
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1, 1, 2, 5, 9, 18, 37, 73, 146, 293, 585, 1170, 2341, 4681, 9362, 18725, 37449, 74898, 149797, 299593, 599186, 1198373, 2396745, 4793490, 9586981, 19173961, 38347922, 76695845, 153391689, 306783378, 613566757, 1227133513, 2454267026, 4908534053, 9817068105
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OFFSET
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0,3
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COMMENTS
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Number of sequences of codewords of total length n from the code C={0,10,110,111}. E.g., a(3)=5 corresponds to the sequences 000, 010, 100, 110 and 111. - Paul Barry, Jan 23 2004
In other words: number of compositions of n into 1 kind of 1's and 2's and two kinds of 3's. - Joerg Arndt, Jun 25 2011
Sums of 3 successive terms are powers of 2. - Mark Dols, Aug 20 2010
For n > 2, a(n) is the number of quaternary sequences of length n (i) starting with q(0)=0; (ii) ending with q(n-1)=0 or 3 and (iii) in which all triples (q(i), q(i+1), q(i+2)) contain digits 0 and 3; cf. A294627. - Wojciech Florek, Jul 30 2018
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REFERENCES
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S. Roman, Introduction to Coding and Information Theory, Springer-Verlag, 1996, p. 42
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LINKS
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FORMULA
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G.f.: 1/((1-2*x)*(1+x+x^2)).
a(n) = a(n-1)+a(n-2)+2*a(n-3). - Paul Curtz, May 23 2008
a(n) = 4*2^n/7 + 3*cos(2*Pi*n/3)/7 + sqrt(3)*sin(2*Pi*n/3)/21. - Paul Barry, Jan 23 2004
Convolution of A000079 and A049347. a(n) = Sum_{k=0..n} 2^k*2*sqrt(3)*cos(2*Pi(n-k)/3+Pi/6)/3. - Paul Barry, May 19 2004
a(n) = sum(sum(binomial(k,j)*binomial(j,n-3*k+2*j)*2^(k-j),j,0,k),k,1,n), n>0. - Vladimir Kruchinin, Sep 07 2010
a(n) = (1/14)*(2^(n + 3) + (-1)^n*((-1)^floor(n/3) + 4*(-1)^floor((n + 1)/3) + 2*(-1)^floor((n + 2)/3) + (-1)^floor((n + 4)/3))). - John M. Campbell, Dec 23 2016
a(n) = (1/63)*(9*2^(2 + n) + (-1)^n*(2 + 9*floor(n/6) - 32*floor((n + 5)/6) + 24*floor((n + 7)/6) + 20*floor((n + 8)/6) - 10*floor((n + 9)/6) - 27*floor((n + 10)/6) + 14*floor((n + 11)/6) + 3*floor((n + 13)/6) - 2*floor((n + 14)/6) + floor((n + 15)/6))). - John M. Campbell, Dec 23 2016
a(n) = T(n+1) + 2*(a(1)*T(n-1) + a(2)*T(n-2) + ... + a(n-2)*T(2) + a(n-1)*T(1)) for T(n) = A000073(n), the tribonacci numbers. - Greg Dresden and Bora Bursalı, Sep 14 2023
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EXAMPLE
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It is shown in A294627 that there are 42 quaternary sequences (i.e. build from four digits 0, 1, 2, 3) and having both 0 and 3 in every (consecutive) triple. Only a(4) = 9 of them start with 0 and end with 0 or 3: 0030, 0033, 0130, 0230, 0300, 0303, 0310, 0320, 0330. - Wojciech Florek, Jul 30 2018
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MAPLE
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MATHEMATICA
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CoefficientList[Series[1/(1 - x - x^2 - 2*x^3), {x, 0, 100}], x] (* or *) LinearRecurrence[{1, 1, 2}, {1, 1, 2}, 70] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
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PROG
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(Maxima) a(n):=sum(sum(binomial(k, j)*binomial(j, n-3*k+2*j)*2^(k-j), j, 0, k), k, 1, n); /* Vladimir Kruchinin, Sep 07 2010 */
(PARI) Vec(1/(1-x-x^2-2*x^3) + O(x^100)) \\ Altug Alkan, Oct 31 2015
(Python)
def A077947(n): return (k:=(m:=1<<n+2)//7) + int((m-7*k<<1)>=7) # Chai Wah Wu, Jan 21 2023
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CROSSREFS
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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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Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021
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STATUS
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approved
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