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A006054
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a(n) = 2*a(n-1) + a(n-2) - a(n-3).
(Formerly M1396)
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48
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0, 0, 1, 2, 5, 11, 25, 56, 126, 283, 636, 1429, 3211, 7215, 16212, 36428, 81853, 183922, 413269, 928607, 2086561, 4688460, 10534874, 23671647, 53189708, 119516189, 268550439, 603427359, 1355888968, 3046654856, 6845771321, 15382308530, 34563733525
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OFFSET
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0,4
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COMMENTS
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Let u(k), v(k), w(k) be 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,... (this sequence with its initial 0 deleted) and {w(n)} = {u(n)} prefixed by an extra 0 = A077998 with an extra initial 0. - Benoit Cloitre, Apr 05 2002. Also u(k)^2+v(k)^2+w(k)^2 = u(2k). - Gary Adamson, Dec 23 2003.
Form the graph with matrix A=[1, 1, 1; 1, 0, 0; 1, 0, 1]. Then A006054 counts walks of length n between the vertex of degree 1 and the vertex of degree 3. - Paul Barry, Oct 02 2004
Form the digraph with matrix [1,1,0;1,0,1;1,1,1]. A006054(n) counts walks of length n between the vertices with loops. - Paul Barry, Oct 15 2004
a(n), n>1 = round(k*A006356(n-1)), where k = .3568958678... = 1/(1+2*Cos Pi/7) - Gary W. Adamson, Jun 06 2008
Nonzero terms = INVERT transform of (1, 1, 2, 2, 3, 3,...). Example: 56 = (1, 1, 2, 5, 11, 25) dot (3, 3, 2, 2, 1, 1) = (3 + 3 + 4 + 10 + 11 + 25). [Gary W. Adamson, Apr 20 2009]
-a(n+1) 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^2-1, the ratio of the larger heptagon diagonal to the side length, as follows. rho^(-n) = C(n)*1+C(n-1)*rho-a(n+1)*sigma, n>=0, with C(n)=A077998(n), C(-1):=0. See the Steinbach reference, and a comment under A052547.
If, with the above notations, the power basis of the field Q(rho) is taken one has for nonpositive powers of rho, rho^(-n) = a(n+2)*1 + A077998(n-1)*rho - a(n+1)*rho^2. For nonnegative powers see A006053. See also the Steinbach reference. - Wolfdieter Lang, May 06 2011.
a(n) appears also in the nonnegative powers of sigma,(defined in the above comment, where also the basis is given). See a comment in A106803.
The sequence b(n):=(-1)^(n+1)*a(n) forms the negative part (i.e. with nonpositive indices) of the sequence (-1)^n*A006053(n+1). In this way we obtain what we shall call the Ramanujan-type sequence number 2a for the argument 2Pi/7 (see the comment to Witula's formula in A006053). We have b(n)=-2*b(n-1)+b(n-2)+b(n-3) and b(n)*49^(1/3) = (c(1)/c(4))^(1/3)*(c(1))^(-n) + (c(2)/c(1))^(1/3)*(c(2))^(-n) + (c(4)/c(2))^(1/3)*(c(4))^(-n) = (c(2)/c(1))^(1/3)*(c(1))^(-n+1) + (c(4)/c(2))^(1/3)*(c(2))^(-n+1) + (c(1)/c(4))^(1/3)*(c(4))^(-n+1), where c(j) := 2*cos(2Pi*j/7) (for the proof see comments to A215112)- Roman Witula, Aug -6 2012.
(1, 1, 2, 5, 11, 25, 56,...) * (1, 0, 1, 0, 1,...) = the variant of A006356: (1, 1, 3, 6, 14, 31,...). [Gary W. Adamson, May 15 2013]
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REFERENCES
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C. P. de Andrade, J. P. de Oliveira Santos, E. V. P. da Silva and K. C. P. Silva, Polynomial Generalizations and Combinatorial Interpretations for Sequences Including the Fibonacci and Pell Numbers, Open Journal of Discrete Mathematics, 2013, 3, 25-32 doi:10.4236/ojdm.2013.31006. - From N. J. A. Sloane, Feb 20 2013
Jay Kappraff, Beyond Measure, A Guided Tour Through Nature, Myth and Number, World Scientific, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
R. Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, J. Integer Seq., 12 (2009), Article 09.8.5.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..150
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 434
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
R. Witula, D. Slota and A. Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
Index to sequences with linear recurrences with constant coefficients, signature (2,1,-1).
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FORMULA
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G.f.: x^2/(1-2*x-x^2+x^3).
Sum_{k, 0<=k<=n+2} a(k) = A077850(n) . - Philippe DELEHAM, Sep 07 2006
Let M = the 3 X 3 matrix [1,1,0; 1,2,1; 0,1,2], then M^n*[1,0,0] = [A080937(n-1), A094790(n), A006054(n-1)]. E.g. M^3*[1,0,0] = [5,9,5] = [A080937(2), A094790(3), A006054(2)]. - Gary W. Adamson, Feb 15 2006
a(n+1) = A187070(2n+1) = A187068(2n+3). - L. Edson Jeffery, Mar 10 2011
a(n+3)=sum(k=1..n, sum(j=0..k, binomial(j,n-3*k+2*j)*(-1)^(j-k)*binomial(k,j)*2^(-n+3*k-j))), a(0)=0, a(1)=0, a(2)=1. [ From Vladimir Kruchinin, May 05 2011]
7*a(n) = (c(2)-c(4))*(1+c(1))^n + (c(4)-c(1))*(1+c(2))^n + (c(1)-c(2))*(1+c(4))^n, where c(j):=2*Cos(2Pi*j/7) - for the proof see Witula et al. papers - Roman Witula, Aug 07 2012.
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MAPLE
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A006054:=z**2/(1-2*z-z**2+z**3); [Simon Plouffe in his 1992 dissertation.]
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MATHEMATICA
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LinearRecurrence[{2, 1, -1}, {0, 0, 1}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)
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PROG
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(Maxima)
a(n):=if n<2 then 0 else if n=2 then 1 else b(n-2);
b(n):=sum(sum(binomial(j, n-3*k+2*j)*(-1)^(j-k)*binomial(k, j)*2^(-n+3*k-j), j, 0, k), k, 1, n);
[Vladimir Kruchinin, May 5 2011]
(Pari) x='x+O('x^66); /* that many terms */
Vec(x^2/(1-2*x-x^2+x^3)) /* show terms */ /* Joerg Arndt, May 5 2011 */
(Haskell)
a006054 n = a006053_list !! n
a006054_list = 0 : 0 : 1 : zipWith (+) (map (2 *) $ drop 2 a006054_list)
(zipWith (-) (tail a006054_list) a006054_list)
-- Reinhard Zumkeller, Oct 14 2011
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CROSSREFS
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Cf. A006356, A007583, A005578, A080937, A094790.
Row sums of A144159 and A180264.
Cf. A006053, A214683, A215112, A214699, A214779.
Sequence in context: A215091 A017919 A017920 * A106805 A094981 A097779
Adjacent sequences: A006051 A006052 A006053 * A006055 A006056 A006057
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KEYWORD
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nonn,easy,changed
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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