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A104934 Expansion of (1-x)/(1 - 3*x - 2*x^2).

%I A104934 #79 May 24 2024 16:04:18
%S A104934 1,2,8,28,100,356,1268,4516,16084,57284,204020,726628,2587924,9217028,
%T A104934 32826932,116914852,416398420,1483024964,5281871732,18811665124,
%U A104934 66998738836,238619546756,849856117940,3026807447332,10780134577876,38394018628292,136742325040628,487015012378468,1734529687216660
%N A104934 Expansion of (1-x)/(1 - 3*x - 2*x^2).
%C A104934 A floretion-generated sequence relating A007482, A007483, A007484. Inverse is A046717. Inverse of Fibonacci(3n+1), A033887. Binomial transform is A052984. Inverse binomial transform is A006131. Note: the conjectured relation 2*a(n) = A007482(n) + A007483(n-1) is a result of the FAMP identity dia[I] + dia[J] + dia[K] = jes + fam
%C A104934 Floretion Algebra Multiplication Program, FAMP Code: 1dia[I]tesseq[A*B] with A = - .25'i + .25'j + .25'k - .25i' + .25j' + .25k' - .25'ii' + .25'jj' + .25'kk' + .25'ij' + .25'ik' + .25'ji' + .25'jk' + .25'ki' + .25'kj' - .25e and B = + 'i + i' + 'ji' + 'ki' + e
%C A104934 a(n) is also the number of ways to build a (2 x 2 x n)-tower using (2 X 1 X 1)-bricks (see Exercise 3.15 in Aigner's book). - Vania Mascioni (vmascioni(AT)bsu.edu), Mar 09 2009
%C A104934 a(n) is the number of compositions of n when there are 2 types of 1 and 4 types of other natural numbers. - _Milan Janjic_, Aug 13 2010
%C A104934 Pisano period lengths: 1, 1, 4, 1, 24, 4, 48, 1, 12, 24, 30, 4, 12, 48, 24, 1,272, 12, 18, 24, ... - _R. J. Mathar_, Aug 10 2012
%D A104934 M. Aigner, A Course in Enumeration, Springer, 2007, p.103.
%H A104934 Vincenzo Librandi, <a href="/A104934/b104934.txt">Table of n, a(n) for n = 0..1000</a>
%H A104934 I. M. Gessel and Ji Li, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Gessel/gessel6.html">Compositions and Fibonacci identities</a>, J. Int. Seq. 16 (2013) 13.4.5
%H A104934 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,2)
%F A104934 Define A007483(-1) = 1. Then 2*a(n) = A007482(n) + A007483(n-1) (conjecture);
%F A104934   a(n+2) = 4*A007484(n) (thus 8*A007484(n) = A007482(n+2) + A007483(n+1));
%F A104934   a(n+1) = 2*A055099(n);
%F A104934   a(n+2) - a(n+1) - a(n) = A007484(n+1) - A007484(n).
%F A104934 a(0)=1, a(1)=2, a(n) = 3*a(n-1) + 2*a(n-2) for n > 1. - _Philippe Deléham_, Sep 19 2006
%F A104934 a(n) = Sum_{k=0..n} 2^k*A122542(n,k). - _Philippe Deléham_, Oct 08 2006
%F A104934 a(n) = ((17+sqrt(17))/34)*((3+sqrt(17))/2)^n + ((17-sqrt(17))/34)*((3-sqrt(17))/2)^n. - _Richard Choulet_, Nov 19 2008
%F A104934 a(n) = 2*a(n-1) + 4*Sum_{k=0..n-2} a(k) for n > 0. - Vania Mascioni (vmascioni(AT)bsu.edu), Mar 09 2009
%F A104934 G.f.: (1-x)/(1-3*x-2*x^2). - _M. F. Hasler_, Jul 12 2018
%F A104934 a(n) = (i*sqrt(2))^(n-1)*( i*sqrt(2)*ChebyshevU(n, -3*i/(2*sqrt(2))) - ChebyshevU(n-1, -3*i/(2*sqrt(2))) ). - _G. C. Greubel_, Jun 27 2021
%F A104934 E.g.f.: exp(3*x/2)*(sqrt(17)*cosh(sqrt(17)*x/2) + sinh(sqrt(17)*x/2))/sqrt(17). - _Stefano Spezia_, May 24 2024
%p A104934 a := proc(n) option remember; `if`(n < 2, [1, 2][n+1], (3*a(n-1) + 2*a(n-2))) end:
%p A104934 seq(a(n), n=0..28); # _Peter Luschny_, Jan 06 2019
%t A104934 LinearRecurrence[{3, 2}, {1, 2}, 40] (* _Vincenzo Librandi_, Jul 13 2018 *)
%t A104934 CoefficientList[Series[(1-x)/(1-3x-2x^2),{x,0,40}],x] (* _Harvey P. Dale_, May 02 2019 *)
%o A104934 (PARI) a(n)=([0,1; 2,3]^n*[1;2])[1,1] \\ _Charles R Greathouse IV_, Jun 20 2015
%o A104934 (Magma) m:=35; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1 - x)/(1 - 3*x - 2*x^2)); // _Vincenzo Librandi_, Jul 13 2018
%o A104934 (Julia) # Following the Pari implementation.
%o A104934 function a(n)
%o A104934    F = BigInt[0 1; 2 3]
%o A104934    Fn = F^n * [1; 2]
%o A104934    Fn[1, 1]
%o A104934 end # _Peter Luschny_, Jan 06 2019
%o A104934 (SageMath) [(i*sqrt(2))^(n-1)*(i*sqrt(2)*chebyshev_U(n, -3*i/(2*sqrt(2))) - chebyshev_U(n-1, -3*i/(2*sqrt(2))) ) for n in (0..40)] # _G. C. Greubel_, Jun 27 2021
%Y A104934 Cf. A006131, A007482 (partial sums), A007483, A007484, A033887, A046717, A052984, A055099, A104935, A122542.
%K A104934 nonn,easy
%O A104934 0,2
%A A104934 _Creighton Dement_, Mar 29 2005