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 A006053 a(n) = a(n-1) + 2*a(n-2) - a(n-3). (Formerly M2358) 41
 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) also appears 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 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 Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 433 László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009. Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992 R. Sachdeva and A. K. Agarwal, Combinatorics of certain restricted n-color composition functions, Discrete Mathematics, 340, (2017), 361-372. Genki Shibukawa, New identities for some symmetric polynomials and their applications, arXiv:1907.00334 [math.CA], 2019. P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31. Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019). 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 + ... Regarding the description "number of compositions of n into floor(j/2) kinds of j's," the a(6)=9 compositions of 6 are (2a,2a,2a), (3a,3a), (2a,4a), (2a,4b), (4a,2a), (4b,2a), (6a), (6b), (6c). - Bridget Tenner, Feb 25 2022 MAPLE a:=0: a:=0: a:=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 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 October 1 16:19 EDT 2022. Contains 357149 sequences. (Running on oeis4.)