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.
Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See pp. 2, 17.
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
E.g.f.: (2*cos(x) + 4*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x))/6. - Stefano Spezia, Nov 09 2024
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
KEYWORD
nonn,easy
AUTHOR
Paul Curtz, Feb 16 2008
EXTENSIONS
More terms from R. J. Mathar, Feb 19 2008
STATUS
approved