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A006053 a(n) = a(n-1) + 2*a(n-2) - a(n-3).
(Formerly M2358)
39
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; text; internal format)
OFFSET

0,5

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, 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

Let c = 2*cos Pi/7 = 1.8019377...; then for n>3, a(n) = 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... - Gary W. Adamson, Jan 24 2010

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. - Wolfdieter Lang, Nov 25 2010

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 + A052547(n-1)*rho + a(n)*rho^2. For negative powers see A006054. - Wolfdieter Lang, May 06 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. - Wolfdieter Lang, Nov 25 2010

REFERENCES

R. Sachdeva and A.K. Agarwal, Combinatorics of certain restricted n-color composition functions, Discrete Mathematics, 340, (2017), 361-372.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. Witula, E. Hetmaniok and D. Slota, Sums of the powers of any order roots taken from the roots of a given polynomial, Proceedings of the 15th International Conference on Fibonacci Numbers and Their Applications (2012)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Robin Chapman and Nicholas C. Singer, Eigenvalues of a bidiagonal matrix, Amer. Math. Monthly, 111 (2004), p. 441

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 433

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.

Roman Witula, Ramanujan Type Trigonometric Formulas: The General Form for the Argument 2Pi/7, J. Integer Seq., 12 (2009), Article 09.8.5.

Index entries for linear recurrences with constant coefficients, signature (1, 2, -1).

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, 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

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, Dec 18 2011

a(n+1)*(-1)^n*49^(1/3) = (c(1)/c(4))^(1/3)*(2*c(1))^n + (c(2)/c(1))^(1/3)*(2*c(2))^n + (c(4)/c(2))^(1/3)*(2c(4))^n = (c(2)/c(1))^(1/3)*(2*c(1))^(n+1) + (c(4)/c(2))^(1/3)*(c(2))^(n+1) + (c(1)/c(4))^(1/3)*(2*c(4))^(n+1), where c(j) := cos(2Pi*j/7); for the proof, see Witula et al.'s papers. - Roman Witula, Jul 21 2012

The previous formula connects the sequence a(n) with A214683, A215076, A215100, A120757. We may call a(n) the Ramanujan-type sequence number 2 for the argument 2*Pi/7. - Roman Witula, Aug 02 2012

a(n) = -A006054(1-n) for all n in Z. - Michael Somos, Nov 30 2014

G.f.: x^2 / (1 - x / (1 - 2*x / (1 + 5*x / (2 - x / (5 - 2*x))))). - Michael Somos, Jan 20 2017

EXAMPLE

G.f. = x^2 + x^3 + 3*x^4 + 4*x^5 + 9*x^6 + 14*x^7 + 28*x^8 + 47*x^9 + ...

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); # Emeric Deutsch

A006053:=z**2/(1-z-2*z**2+z**3); # conjectured by Simon Plouffe in his 1992 dissertation

MATHEMATICA

LinearRecurrence[{1, 2, -1}, {0, 0, 1}, 50]  (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)

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

(PARI) {a(n) = if( n<0, n = -1-n; polcoeff( -1 / (1 - 2*x - x^2 + x^3) + x * O(x^n), n), polcoeff( x^2 / (1 - x - 2*x^2 + x^3) + x * O(x^n), n))}; /* Michael Somos, Nov 30 2014 */

CROSSREFS

Cf. A006054, A096975, A096976.

Sequence in context: A014596 A002823 A109509 * A051841 A096081 A054162

Adjacent sequences:  A006050 A006051 A006052 * A006054 A006055 A006056

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Emeric Deutsch, Dec 14 2004

Typo in definition fixed by Reinhard Zumkeller, Oct 14 2011

STATUS

approved

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Last modified May 29 16:53 EDT 2017. Contains 287252 sequences.