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 A135318 The Kentucky-2 sequence: a(n) = a(n-2) + 2*a(n-4), with a[0..3] = [1, 1, 1, 2]. 9
 1, 1, 1, 2, 3, 4, 5, 8, 11, 16, 21, 32, 43, 64, 85, 128, 171, 256, 341, 512, 683, 1024, 1365, 2048, 2731, 4096, 5461, 8192, 10923, 16384, 21845, 32768, 43691, 65536, 87381, 131072, 174763, 262144, 349525, 524288, 699051, 1048576, 1398101, 2097152, 2796203 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Shifted Jacobsthal recurrence. From L. Edson Jeffery, Apr 21 2011: (Start) Let U be the unit-primitive matrix (see [Jeffery]) U=U_(6,2)= (0 0 1) (0 2 0) (2 0 1), let i in {0,1}, m>=0 an integer and n=2*m+i. Then a(n)=a(2*m+i)=Sum_{j=0..2} (U^m)_(i,j). (End) a(n) is also the pebbling number of the cycle graph C_{n+1} for n > 1. - Eric W. Weisstein, Jan 07 2021 From Greg Dresden and Ziyi Xie, Aug 25 2023: (Start) a(n) is the number of ways to tile a zig-zag strip of n cells using squares (of 1 cell) and triangles (of 3 cells). Here is the zig-zag strip corresponding to n=11, with 11 cells: ___ ___ ___| |___| |___ | |___| |___| |___ |___| |___| |___| | | |___| |___| |___| |___| |___| |___|, and here are the two types of triangles (where one is just a reflection of the other): ___ ___ | |___ ___| | | | | | | ___| and |___ | |___| |___|. As an example, here is one of the a(11) = 32 ways to tile the zig-zag strip of 11 cells: ___ ___ ___| |___| |___ | |___| | |___ | |___ | | | ___| |___| ___| |___| |___| |___|. (End) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..5000 Minerva Catral et al., Generalizing Zeckendorf's Theorem: The Kentucky Sequence, arXiv:1409.0488 [math.NT], 2014. See 1.3 p. 2, same sequence without the first 2 terms. L. E. Jeffery, Unit-primitive matrices. Eric Weisstein's World of Mathematics, Cycle Graph Eric Weisstein's World of Mathematics, Pebbling Number Index entries for linear recurrences with constant coefficients, signature (0,1,0,2). FORMULA From R. J. Mathar, Feb 19 2008: (Start) O.g.f.: [1/(1+x^2)+(-2-3*x)/(2*x^2-1)]/3. a(2n) = A001045(n+1). a(2n+1) = A000079(n). (End) From L. Edson Jeffery, Apr 21 2011: (Start) G.f.: (1+x+x^3)/((1+x^2)*(1-2*x^2)). a(n) = (((-i)^(n+1)-i^(n+1))*2*i*sqrt(2)+3*(1+(-1)^(n+1))*2^((n+2)/2)+(1-(-1)^(n+1))*2^((n+5)/2))/(12*sqrt(2)), where i=sqrt(-1). (End) a(n) = (2^floor(n/2)*(5-(-1)^n)+(-1)^floor(n/2)*(1+(-1)^n))/6 = (A016116(n)*A010711(n)+2*A056594(n))/6. - Bruno Berselli, Apr 21 2011 a(2n) = 2*a(2n-1) - a(2n-2); a(2n+1) = a(2n) + a(2n-2). - Richard R. Forberg, Aug 19 2013 a(n) = A112387(n + (-1)^n). - Alois P. Heinz, Sep 28 2023 EXAMPLE Let i=0 and m=3. Then U^3 = (2,0,3;0,8,0;6,0,5), and the first-row sum (corresponding to i=0) is 2 + 0 + 3 = 5. Hence a(n) = a(2*m+i) = a(2*3+0) = a(6) = 2 + 3 = 5. MAPLE a:= n-> (<<0|1>, <2|1>>^(iquo(n, 2, 'm')). <<1, 1+m>>)[1, 1]: seq(a(n), n=0..50); # Alois P. Heinz, May 30 2022 MATHEMATICA LinearRecurrence[{0, 1, 0, 2}, {1, 1, 1, 2}, 40] (* Harvey P. Dale, Oct 14 2015 *) PROG (Magma) [(2^Floor(n/2)*(5-(-1)^n)+(-1)^Floor(n/2)*(1+(-1)^n))/6: n in [0..50]]; // Vincenzo Librandi, Aug 10 2011 CROSSREFS Cf. A000079, A001045, A112387. Sequence in context: A302592 A078762 A103262 * A374782 A210671 A189761 Adjacent sequences: A135315 A135316 A135317 * A135319 A135320 A135321 KEYWORD nonn,easy AUTHOR Paul Curtz, Feb 16 2008 EXTENSIONS More terms from R. J. Mathar, Feb 19 2008 STATUS approved

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Last modified August 11 23:45 EDT 2024. Contains 375082 sequences. (Running on oeis4.)