

A077998


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


26



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)=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
Diagonal sums of A056242.  Paul Barry, Dec 26 2007
Diagonal sums of triangle in A105306.  Philippe Deléham, Nov 16 2008
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


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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 2231.
Alexey Ustinov, Supercontinuants, arXiv:1503.04497 [math.NT], 2015.
R. Witula, D. Slota and A. Warzynski, QuasiFibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index entries for linear recurrences with constant coefficients, signature (2,1,1).


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
a(n) = b(n+2) b(n+1), first differences of b(n) = A006054(n).  Wolfdieter Lang, Nov 07 2013; corrected by Kai Wang, May 31 2017


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 *)


PROG

(PARI) a(n)=([0, 1, 0; 0, 0, 1; 1, 1, 2]^n*[1; 1; 3])[1, 1] \\ Charles R Greathouse IV, May 10 2016
(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
(Sage) ((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.
Cf. A105477.
Sequence in context: A106803 A199853 A006356 * A209357 A090165 A129954
Adjacent sequences: A077995 A077996 A077997 * A077999 A078000 A078001


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Nov 17 2002


EXTENSIONS

Edited by N. J. A. Sloane, Aug 08 2008 at the suggestion of R. J. Mathar


STATUS

approved



