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A066983 a(1)=a(2)=1; for n >= 1, a(n+2) = a(n+1) + a(n) + (-1)^n. 16
1, 1, 1, 3, 3, 7, 9, 17, 25, 43, 67, 111, 177, 289, 465, 755, 1219, 1975, 3193, 5169, 8361, 13531, 21891, 35423, 57313, 92737, 150049, 242787, 392835, 635623, 1028457, 1664081, 2692537, 4356619, 7049155, 11405775, 18454929, 29860705 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Length of strings given by a successive substitution of a "modified" Kolakoski-(3, 1) sequence. Starting with 1, using the rule "string begins with 1 if previous string ends with 3, string begins with 3 if previous string ends with 1" then applying the classical Kolakoski-(3,1) rule. This gives: 1 -> 3 -> 111 -> 313 -> 1113111 -> 313111313 -> 11131113131113111 and the length of string are 1, 1, 3, 3, 7, 9, 17, ... At step n, length = a(n+1). This substitution leads to two sequences: 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, ... and 3, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, ... - Benoit Cloitre, Jun 01 2004

Lengths of comparators in subsequent layers of correction network F_n. - Grzegorz Stachowiak (gst(AT)ii.uni.wroc.pl), Nov 28 2004

Convolution of F(n+1) and A105812(n). Action of inverse of sequence array for F(n-1)*(-1)^n on F(n+1). - Paul Barry, Oct 29 2006

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..250

K. Atanassov, D. Dimitrov and A. G. Shannon, A remark on psi-function and Pell-Padovan's sequence, Notes Number Theory Discrete Math., 15 (2009), no. 2, 1-44.

Michael Baake and Bernd Sing, Kolakoski-(3,1) is a (deformed) model set, arXiv:math/0206098 [math.MG], 2002-2003.

G. Stachowiak, Fibonacci Correction Networks, SWAT 2000, LNCS 1851, 535-548.

G. Stachowiak, Lower Bounds on Correction Networks, ISAAC 2003, LNCS 2906, 221-229.

Dursun Tasci, Gaussian Padovan and Gaussian Pell-Padovan sequences, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 67 (2018), no. 2, 82-88. Sequence R_n.

Index entries for linear recurrences with constant coefficients, signature (0,2,1)

FORMULA

For n > 4 a(n-2) = floor(2 * phi^n/sqrt(5)) + (1 + (-1)^n)/2

a(n) = 2 * Fibonacci(n-2) + (-1)^n. - Vladeta Jovovic, Mar 19 2003

G.f.: x(1+x-x^2)/((1+x)(1-x-x^2)). - Paul Barry, Oct 29 2006

a(n) = A066629(n-2) - A066629(n-3), n > 2. - R. J. Mathar, Jan 14 2009

a(n) = floor(phi^(n-1)) - floor(phi^(n-1)/sqrt(5)). - Federico Provvedi, Mar 26 2013

a(0) = a(1) = a(2) = 1; for n > 2, a(n) = 2*a(n-2) + a(n-3). - Taras Goy, Aug 03 2018

MAPLE

seq(coeff(series(x*(1+x-x^2)/((1+x)*(1-x-x^2)), x, n+1), x, n), n=1..40); # Muniru A Asiru, Aug 09 2018

MATHEMATICA

Table[ Floor[ GoldenRatio^(k-1) ] - Floor[ GoldenRatio^(k-1) / Sqrt[5] ], {k, 1, 100} ]  (* Federico Provvedi, Mar 26 2013 *)

LinearRecurrence[{0, 2, 1}, {1, 1, 1}, 40] (* Vincenzo Librandi, Aug 13 2018 *)

PROG

(PARI) { for (n=1, 250, if (n>2, a=a1 + a2 + (-1)^n; a2=a1; a1=a, a=a1=1; a=a2=1); write("b066983.txt", n, " ", a) ) } \\ Harry J. Smith, Apr 15 2010

(GAP) a:=[1, 1];; for n in [3..40] do a[n]:=a[n-1]+a[n-2]+(-1)^n; od; a; # Muniru A Asiru, Aug 09 2018

(MAGMA) [n le 2 select 1 else Self(n-1)+Self(n-2)+(-1)^n: n in [1..50]]; // Vincenzo Librandi, Aug 13 2018

CROSSREFS

Cf. A064353, A001083, A042942.

Sequence in context: A285187 A034411 A258289 * A048240 A122012 A185306

Adjacent sequences:  A066980 A066981 A066982 * A066984 A066985 A066986

KEYWORD

nonn,easy

AUTHOR

Benoit Cloitre, Jan 27 2002

STATUS

approved

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Last modified April 25 12:19 EDT 2019. Contains 322456 sequences. (Running on oeis4.)