cp's OEIS Frontend

A077947 Expansion of 1/(1 - x - x^2 - 2*x^3).

%I A077947 #143 Sep 15 2023 07:52:43
%S A077947 1,1,2,5,9,18,37,73,146,293,585,1170,2341,4681,9362,18725,37449,74898,
%T A077947 149797,299593,599186,1198373,2396745,4793490,9586981,19173961,
%U A077947 38347922,76695845,153391689,306783378,613566757,1227133513,2454267026,4908534053,9817068105
%N A077947 Expansion of 1/(1 - x - x^2 - 2*x^3).
%C A077947 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
%C A077947 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
%C A077947 Diagonal sums of number Pascal-(1,2,1) triangle A081577. - _Paul Barry_, Jan 24 2005
%C A077947 For n>0: a(n) = A173593(2*n+1) - A173593(2*n); a(n+1) = A173593(2*n) - A173593(2*n-1). - _Reinhard Zumkeller_, Feb 22 2010
%C A077947 Sums of 3 successive terms are powers of 2. - _Mark Dols_, Aug 20 2010
%C A077947 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
%D A077947 S. Roman, Introduction to Coding and Information Theory, Springer-Verlag, 1996, p. 42
%H A077947 Vincenzo Librandi, <a href="/A077947/b077947.txt">Table of n, a(n) for n = 0..1000</a>
%H A077947 G. Cerda-Morales, <a href="https://arxiv.org/abs/1905.00725">A note on modified third-order Jacobsthal Numbers</a>, arxiv:1905.00725 [math.CO], 2019.
%H A077947 M. H. Cilasun, <a href="http://arxiv.org/abs/1412.3265">An Analytical Approach to Exponent-Restricted Multiple Counting Sequences</a>, arXiv preprint arXiv:1412.3265 [math.NT], 2014.
%H A077947 M. H. Cilasun, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Cilasun/cila5.html">Generalized Multiple Counting Jacobsthal Sequences of Fermat Pseudoprimes</a>, Journal of Integer Sequences, Vol. 19, 2016, #16.2.3.
%H A077947 Charles K. Cook and Michael R. Bacon, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from27to39.pdf">Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations</a>, Annales Mathematicae et Informaticae, 41 (2013) pp. 27-39.
%H A077947 W. Florek, <a href="http://doi.org/10.1016/j.amc.2018.06.014">A class of generalized Tribonacci sequences applied to counting problems</a>, Appl. Math. Comput., 338 (2018), 809-821.
%H A077947 S. Kitaev, J. Remmel and M. Tiefenbruck, <a href="http://arxiv.org/abs/1201.6243">Marked mesh patterns in 132-avoiding permutations I</a>, arXiv preprint arXiv:1201.6243 [math.CO], 2012. - From _N. J. A. Sloane_, May 09 2012
%H A077947 Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, <a href="http://www.emis.de/journals/INTEGERS/papers/p16/p16.Abstract.html">Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II</a>, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16; see also <a href="http://arxiv.org/abs/1302.2274">arXiv preprint</a>, arXiv:1302.2274 [math.CO], 2013.
%H A077947 Vladimir Victorovich Kruchinin, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565, arXiv:1009.2565 [math.CO], 2010.
%H A077947 Evren Eyican Polatlı and Yüksel Soykan, <a href="https://doi.org/10.9734/ARJOM/2021/v17i230270">On generalized third-order Jacobsthal numbers</a>, Asian Res. J. of Math. (2021) Vol. 17, No. 2, 1-19, Article No. ARJOM.66022.
%H A077947 Yüksel Soykan, <a href="https://arxiv.org/abs/1910.03490">Summing Formulas For Generalized Tribonacci Numbers</a>, arXiv:1910.03490 [math.GM], 2019.
%H A077947 Anthony Zaleski and Doron Zeilberger, <a href="https://arxiv.org/abs/1712.10072">On the Intriguing Problem of Counting (n+1,n+2)-Core Partitions into Odd Parts</a>, arXiv:1712.10072 [math.CO], 2017.
%H A077947 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,2).
%F A077947 G.f.: 1/((1-2*x)*(1+x+x^2)).
%F A077947 a(n) = a(n-1)+a(n-2)+2*a(n-3). - _Paul Curtz_, May 23 2008
%F A077947 a(n) = round(2^(n+2)/7). - _Mircea Merca_, Dec 28 2010
%F A077947 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
%F A077947 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
%F A077947 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
%F A077947 Partial sums of A078010 starting (1, 0, 1, 3, 4, 9, ...). - _Gary W. Adamson_, May 13 2013
%F A077947 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
%F A077947 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
%F A077947 7*a(n) = 2^(n+2) + A167373(n+1). - _R. J. Mathar_, Feb 06 2020
%F A077947 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
%e A077947 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
%p A077947 seq(round(2^(n+2)/7),n=0..25); # _Mircea Merca_, Dec 28 2010
%t A077947 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 *)
%o A077947 (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 */
%o A077947 (Magma) [Round(2^(n+2)/7): n in [0..40]]; // _Vincenzo Librandi_, Jun 25 2011
%o A077947 (PARI) Vec(1/(1-x-x^2-2*x^3) + O(x^100)) \\ _Altug Alkan_, Oct 31 2015
%o A077947 (Python)
%o A077947 def A077947(n): return (k:=(m:=1<<n+2)//7) + int((m-7*k<<1)>=7) # _Chai Wah Wu_, Jan 21 2023
%Y A077947 Apart from signs, same as A077972.
%Y A077947 Cf. A139217 and A139218.
%Y A077947 Cf. A078010.
%Y A077947 Cf. A294627.
%K A077947 nonn,easy
%O A077947 0,3
%A A077947 _N. J. A. Sloane_, Nov 17 2002
%E A077947 Deleted certain dangerous or potentially dangerous links. - _N. J. A. Sloane_, Jan 30 2021