

A077998


Expansion of (1x)/(12*xx^2+x^3).


32



1, 1, 3, 6, 14, 31, 70, 157, 353, 793, 1782, 4004, 8997, 20216, 45425, 102069, 229347, 515338, 1157954, 2601899, 5846414, 13136773, 29518061, 66326481, 149034250, 334876920, 752461609, 1690765888, 3799116465, 8536537209, 19181424995, 43100270734, 96845429254
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OFFSET

0,3


COMMENTS

Let u(k), v(k), w(k) be defined by u(1)=1, v(1)=0, w(1)=0 and u(k+1)=u(k)+v(k)+w(k), v(k+1)=u(k)+v(k), w(k+1)=u(k); then {u(n)} = 1,1,3,6,14,31,... (A006356 with an extra initial 1), {v(n)} = 0,1,2,5,11,25,... (A006054 with its initial 0 deleted) and {w(n)} = {u(n)} prefixed by an extra 0 = this sequence with an extra initial 0.  Benoit Cloitre, Apr 05 2002 [Also u(k)^2+v(k)^2+w(k)^2 = u(2k).  Gary W. Adamson, Dec 23 2003]
Form the graph with matrix A=[1, 1, 1; 1, 0, 0; 1, 0, 1]. Then A077998 counts closed walks of length n at the vertex of degree 4.  Paul Barry, Oct 02 2004
a(n) is the number of Motzkin (n+2)sequences with no flatsteps at ground level and whose height is <=2. For example, a(3)=6 counts UDUFD, UFDUD, UFFFD, UFUDD, UUDFD, UUFDD.  David Callan, Dec 09 2004
Number of compositions of n if there are two kinds of part 2. Example: a(3)=6 because we have (3),(1,2),(1,2'),(2,1),(2',1) and (1,1,1). Row sums of A105477.  Emeric Deutsch, Apr 09 2005
a(n) appears in the formula for the nonpositive powers of rho:= 2*cos(Pi/7), the ratio of the smaller diagonal in the heptagon to the side length s=2*sin(Pi/7), when expressed in the basis <1,rho,sigma>, with sigma:=rho^21, the ratio of the larger heptagon diagonal to the side length, as follows. rho^(n) = a(n)*1 + a(n1)*rho  C(n)*sigma, n>=0, with C(n)=A006054(n+1). Put a(1):=0. See the Steinbach reference, and a comment under A052547.
The limit a(n+1)/a(n) for n > infinity is sigma = rho^21, approximately 2.246979603. See a Nov 07 2013 comment on A006054 for the proof, and the preceding comment for rho and sigma and the P. Steinbach reference.  Wolfdieter Lang, Nov 07 2013
a(n) is the number of ways to tile a skew doublestrip of 3*n cells using all possible "trominos". Here is the skew doublestrip corresponding to n=4, with 12 cells:
___ ___ ___ ___ ___ ___
      
______________ ____
      
__________________,
and here are the three possible "tromino" tiles, which can be rotated or reflected as needed:
___ ___
   
_____ ________ ___________
         
______, ______ , _________.
As an example, here is one of the a(4) = 14 ways to tile the skew doublestrip of 12 cells:
___ ___ _______ _______
    
_ _ _____ _ _
    
____________________. (End)


REFERENCES

Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169177.
Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.


LINKS



FORMULA

a(0)=a(1)=1, a(2)=3, a(n+1) = 2*a(n) + a(n1)  a(n2) for n>=2.  Philippe Deléham, Sep 07 2006
7*a(n) = (s(2))^2*(1+c(1))^n + (s(4))^2*(1+c(2))^n + (s(1))^2(1+c(4))^n, where c(j) = 2*Cos(2Pi*j/7) and s(j) = 2*Sin(2Pi*j/7)  for the proof of this one and many other relations for the sequences u(k), v(k) and w(k) defined on the top of the comments by Benoit Cloitre  see Witula et al.'s paper.  Roman Witula, Aug 07 2012


EXAMPLE

G.f. = 1 + x + 3*x^2 + 6*x^3 + 14*x^4 + 31*x^5 + 70*x^6 + 157*x^7 + 353*x^8 + ...  Michael Somos, Dec 12 2023


MATHEMATICA

CoefficientList[Series[(1x)/(12*xx^2+x^3), {x, 0, 40}], x] (* Stefan Steinerberger, Sep 11 2006 *)
LinearRecurrence[{2, 1, 1}, {1, 1, 3}, 40] (* Roman Witula, Aug 07 2012 *)
a[ n_] := {1, 0, 0} . MatrixPower[{{0, 1, 0}, {0, 0, 1}, {1, 1, 2}}, n] . {1, 1, 3}; (* Michael Somos, Dec 12 2023 *)


PROG

(Magma) I:=[1, 1, 3]; [n le 3 select I[n] else 2*Self(n1)+Self(n2)Self(n3): n in [1..40]]; // Vincenzo Librandi, Jun 01 2017
(SageMath) ((1x)/(12*xx^2+x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 27 2019
(GAP) a:=[1, 1, 3];; for n in [4..40] do a[n]:=2*a[n1]+a[n2]a[n3]; od; a; # G. C. Greubel, Jun 27 2019


CROSSREFS

Apart from initial term, same as A006356, which is the main entry for this sequence. A106803 is yet another version.


KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



