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

1, 2, 8, 28, 100, 356, 1268, 4516, 16084, 57284, 204020, 726628, 2587924, 9217028, 32826932, 116914852, 416398420, 1483024964, 5281871732, 18811665124, 66998738836, 238619546756, 849856117940, 3026807447332, 10780134577876, 38394018628292, 136742325040628, 487015012378468, 1734529687216660

Offset: 0

Author

Creighton Dement, Mar 29 2005

Keywords

• nonn
• easy

Comments

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

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

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

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

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

References

• M. Aigner, A Course in Enumeration, Springer, 2007, p.103.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
• Index entries for linear recurrences with constant coefficients, signature (3,2)

Crossrefs

Cf. A006131, A007482 (partial sums), A007483, A007484, A033887, A046717, A052984, A055099, A104935, A122542.

Programs

Julia

# Following the Pari implementation.
function a(n)
   F = BigInt[0 1; 2 3]
   Fn = F^n * [1; 2]
   Fn[1, 1]
end # Peter Luschny, Jan 06 2019

Magma

m:=35; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1 - x)/(1 - 3*x - 2*x^2)); // Vincenzo Librandi, Jul 13 2018

Maple

a := proc(n) option remember; `if`(n < 2, [1, 2][n+1], (3*a(n-1) + 2*a(n-2))) end:
seq(a(n), n=0..28); # Peter Luschny, Jan 06 2019

Mathematica

LinearRecurrence[{3, 2}, {1, 2}, 40] (* Vincenzo Librandi, Jul 13 2018 *)
CoefficientList[Series[(1-x)/(1-3x-2x^2),{x,0,40}],x] (* Harvey P. Dale, May 02 2019 *)

PARI

a(n)=([0,1; 2,3]^n*[1;2])[1,1] \\ Charles R Greathouse IV, Jun 20 2015

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

Formula

Define A007483(-1) = 1. Then 2*a(n) = A007482(n) + A007483(n-1) (conjecture);

a(n+2) = 4*A007484(n) (thus 8*A007484(n) = A007482(n+2) + A007483(n+1));

a(n+1) = 2*A055099(n);

a(n+2) - a(n+1) - a(n) = A007484(n+1) - A007484(n).

a(0)=1, a(1)=2, a(n) = 3*a(n-1) + 2*a(n-2) for n > 1. - Philippe Deléham, Sep 19 2006

a(n) = Sum_{k=0..n} 2^k*A122542(n,k). - Philippe Deléham, Oct 08 2006

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

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

G.f.: (1-x)/(1-3*x-2*x^2). - M. F. Hasler, Jul 12 2018

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

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