A066983 a(n+2) = a(n+1) + a(n) + (-1)^n, with a(1) = a(2) = 1.

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, 48315633, 78176339

Offset: 1

Author

Benoit Cloitre, Jan 27 2002

Keywords

• nonn
• easy

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

References

• Omur Deveci, The Pell-Padovan sequences and the Jacobsthal-Padovan sequences in finite groups, Utilitas Mathematica, 98 (2015), 257-270.

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.
• Taras Goy, S. V. Sharyn, A note on Pell-Padovan numbers and their connection with Fibonacci numbers, Carpathian Math. Publ. (2020) Vol. 12, No. 2, 280-288.
• Zehra İşbilir and Nurten Gürses, Pell-Padovan generalized quaternions, Notes on Num. Theory and Disc. Math. (2021) Vol. 27, No. 1, 171—187.
• 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).

Crossrefs

Cf. A000045, A001083, A042942, A064353, A066629.

Programs

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

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 *)

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

PARI

vector(40, n, 2*fibonacci(n-2) + (-1)^n) \\ G. C. Greubel, Dec 26 2019

Python

from sympy import fibonacci
def A066983(n): return (fibonacci(n-2)<<1)+(-1 if n&1 else 1) # Chai Wah Wu, May 05 2025

Sage

[2*fibonacci(n-2) + (-1)^n for n in (1..40)] # G. C. Greubel, Dec 26 2019

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

a(n) = (-1)^n + (-1 - 3/sqrt(5))*((1/2)*(1 - sqrt(5)))^n + (-1 + 3/sqrt(5))*((1/2)*(1 + sqrt(5)))^n. - Stefano Spezia, Jul 22 2019

Extensions

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021