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A135318 a(n) = a(n-2) + 2*a(n-4), with a[0..3] = [1, 1, 1, 2]. 2
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 (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

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 NumberIndex 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

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.

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

Sequence in context: A302592 A078762 A103262 * A210671 A189761 A329576

Adjacent sequences:  A135315 A135316 A135317 * A135319 A135320 A135321

KEYWORD

nonn

AUTHOR

Paul Curtz, Feb 16 2008

EXTENSIONS

More terms from R. J. Mathar, Feb 19 2008

STATUS

approved

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Last modified February 26 03:00 EST 2021. Contains 341619 sequences. (Running on oeis4.)