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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)

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.

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

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 November 26 21:20 EST 2020. Contains 338641 sequences. (Running on oeis4.)