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A006053
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a(n) = a(n-1) + 2*a(n-2) - a(n-3).
(Formerly M2358)
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25
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0, 0, 1, 1, 3, 4, 9, 14, 28, 47, 89, 155, 286, 507, 924, 1652, 2993, 5373, 9707, 17460, 31501, 56714, 102256, 184183, 331981, 598091, 1077870, 1942071, 3499720, 6305992, 11363361, 20475625, 36896355, 66484244, 119801329, 215873462, 388991876, 700937471
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| a(n+1)=S(n) for n>=1, where S(n) is the number of 01-words of length n, having first letter 1, in which all runlengths of 1's are odd. Example: S(4) counts 1000,1001,1010,1110. See A077865. - Clark Kimberling (ck6(AT)evansville.edu), Jun 26 2004
For n>=1, number of compositions of n into floor(j/2) kinds of j's (see g.f.). [Joerg Arndt, Jul 06 2011]
Counts walks of length n between the first and second nodes of P_3, to which a loop has been added at the end. Let A be the adjacency matrix of the graph P_3 with a loop added at the end. A is a 'reverse Jordan matrix' [0,0,1;0,1,1;1,1,0]. a(n) is obtained by taking the (1,2) element of A^n. - Paul Barry, Jul 16 2004
Interleaves A094790 and A094789. - Paul Barry, Oct 30 2004
Contribution from Gary W. Adamson, Jan 24 2010: (Start)
Let c = 2*Cos Pi/7 = 1.8019377...; then a(n), n>3 = c^(n-2) - a(n-1)*(c-1) +
(1/c)*a(n-2). Example: a(7) = 14 = c^5 - 9*(c-1) + 4/c = 18.997607...
- 7.19806226... + 2.219832528... (End)
a(n) appears in the formula for the nonnegative 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)*sigma, n>=0, with C(n)=A052547(n-2). See the Steinbach reference, and a comment under A052547.
If with the above notations the power basis <1,rho,rho^2> of Q(rho) is used, nonnegative powers of rho are given by rho^n = -a(n-1)*1 + A0052547(n-1)*rho + a(n)*rho^2. For negative powers see A006054. Wolfdieter Lang, May 6 2011.
-a(n-1) appears also in the formula for the nonpositive powers of sigma (see the above comment for the definition, and the Steinbach basis <1,rho,sigma>) as follows. sigma^(-n) = A(n)*1 -a(n+1)*rho -A(n-1)*sigma, with A(n)=A052547(n), A(-1):=0.
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REFERENCES
| R. Chapman, Eigenvalues of a Bidiagonal Matrix, Amer. Math. Monthly, 111 (2004) p. 441
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, Mathematics Magazine, 70,1 (1997) 22-31.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
S. 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.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 433
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FORMULA
| G.f.: x^2/(1-x-2*x^2+x^3). - Emeric Deutsch, Dec 14 2004
G.f.: -1 + 1/(1-sum(j>=1, floor(j/2)*x^j )). [Joerg Arndt, Jul 06 2011]
a(n+2)=A094790(n/2+1)(1+(-1)^n)/2+A094789((n+1)/2)(1-(-1)^n)/2 - Paul Barry, Oct 30 2004
First differences of A028495. - Floor van Lamoen (fvlamoen(AT)hotmail.com), Nov 02 2005
a(n)=A187065(2*n+1); a(n+1)=A187066(2*n+1)=A187067(2*n). - L. Edson Jeffery, Mar 16, 2011.
G.f.: x^2/(1-x-2*x^2+x^3). - Emeric Deutsch, Dec 14 2004
With c[j_]:=Cos[Pi*j/7] we have
a[n_]:= 2^n*( c[1]^(n-1)*(c[1]+ c[2]) + c[3]^(n-1)*(c[3]+c[6]) + c[5]^(n-1)*(c[5]+c[4]) )/7 - Herbert Kociemba (kociemba(AT)t-online.de), Dec 18 2011
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MAPLE
| a[0]:=0: a[1]:=0: a[2]:=1: for n from 3 to 40 do a[n]:=a[n-1]+2*a[n-2]-a[n-3] od:seq(a[n], n=0..40); (Deutsch)
A006053:=z**2/(1-z-2*z**2+z**3); [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| LinearRecurrence[{1, 2, -1}, {0, 0, 1}, 50] (* From Vladimir Joseph Stephan Orlovsky, May 25 2011 *)
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PROG
| (MAGMA) [ n eq 1 select 0 else n eq 2 select 0 else n eq 3 select 1 else Self(n-1)+2*Self(n-2)-Self(n-3): n in [1..40] ]: // Vincenzo Librandi, Aug 19 2011
(Haskell)
a006053 n = a006053_list !! n
a006053_list = 0 : 0 : 1 : zipWith (+) (drop 2 a006053_list)
(zipWith (-) (map (2 *) $ tail a006053_list) a006053_list)
-- Reinhard Zumkeller, Oct 14 2011
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CROSSREFS
| Cf. A096975, A096976.
Sequence in context: A014596 A002823 A109509 * A051841 A096081 A054162
Adjacent sequences: A006050 A006051 A006052 * A006054 A006055 A006056
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 14 2004
Typo in definition fixed
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