A189050 a(n) = if n even then P((n-2)/2)+P(n/2) otherwise 3*P((n+1)/2)+P((n-1)/2) where P(i) are the Pell numbers (A000129).

3, 1, 7, 3, 17, 7, 41, 17, 99, 41, 239, 99, 577, 239, 1393, 577, 3363, 1393, 8119, 3363, 19601, 8119, 47321, 19601, 114243, 47321, 275807, 114243, 665857, 275807, 1607521, 665857, 3880899, 1607521, 9369319, 3880899, 22619537, 9369319, 54608393, 22619537, 131836323, 54608393, 318281039, 131836323, 768398401, 318281039, 1855077841, 768398401

Offset

1,1

References

R. P. Grimaldi, Ternary strings with no consecutive 0's and no consecutive 1's, Congressus Numerantium, 205 (2011), 129-149. See p. 142.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

Formula

a(n) = 2*a(n-2)+a(n-4). G.f.: -x*(3+x+x^2+x^3) / ( -1+2*x^2+x^4 ). - Colin Barker, Jul 24 2013

a(n) = a(n-1)+2*a(n-2) if n odd. a(n) =(a(n-1)-a(n-2))/2 if n even. - R. J. Mathar, Jun 18 2014

Maple

A000129 := proc(n) option remember; if n <=1 then n; else 2*A000129(n-1)+A000129(n-2); fi; end; P:=A000129;
pal:=n-> if n mod 2 = 0 then P((n-2)/2)+P(n/2) else 3*P((n+1)/2)+P((n-1)/2); fi;

Mathematica

nxt[{n_, a_, b_}]:={n+1, b, If[EvenQ[n], b+2a, (b-a)/2]}; NestList[nxt, {2, 3, 1}, 50][[All, 2]] (* or *) LinearRecurrence[{0, 2, 0, 1}, {3, 1, 7, 3}, 50] (* Harvey P. Dale, Mar 06 2019 *)

Crossrefs

Cf. A000129 (Pell numbers), A001333 (bisections)

Sequence in context: A333847 A342268 A316742 * A095868 A140962 A013602

Adjacent sequences: A189047 A189048 A189049 * A189051 A189052 A189053

Keyword

nonn,easy

Author

N. J. A. Sloane, Apr 16 2011

Status

approved