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A078343 a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2). 18
-1, 2, 3, 8, 19, 46, 111, 268, 647, 1562, 3771, 9104, 21979, 53062, 128103, 309268, 746639, 1802546, 4351731, 10506008, 25363747, 61233502, 147830751, 356895004, 861620759, 2080136522, 5021893803, 12123924128, 29269742059, 70663408246, 170596558551 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Inverse binomial transform of -1, 1, 6, 22, 76, 260,.. (see A111566). Binomial transform of -1, 3, -2, 6, -4, 12, -8, 24, -16,... (see A162255). - R. J. Mathar, Oct 02 2012

REFERENCES

H. S. M. Coxeter, 1998, Numerical distances among the circles in a loxodromic sequence, Nieuw Arch. Wisk, 16, pp. 1-9.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Tanya Khovanova, Recursive Sequences

José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368, 2012

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

FORMULA

For the unsigned version: a(1)=1; a(2)=2; a(n) = sum_{k=2..n-1} (a(k) + a(k-1)).

a(n) is asymptotic to (1/4)*(8-5*sqrt(2))*(1+sqrt(2))^n.

a(n) = A048746(n-3) + 2, for n>2. - Ralf Stephan, Oct 17 2003

a(n) = 2*A000129(n) - A000129(n-1) if n>0; abs(a(n)) = sum_{k=0..floor(n/2)} (C(n-k-1, k) - C(n-k-1, k-1))2^(n-2k). - Paul Barry, Dec 23 2004

O.g.f.: (1-4*x)/(-1 + 2*x + x^2). - R. J. Mathar, Feb 15 2008

a(n) = 4*Pell(n) - Pell(n+1), where Pell = A000129. - Vladimir Reshetnikov, Sep 27 2016

MAPLE

f:=proc(n) option remember; if n=0 then RETURN(-1); fi; if n=1 then RETURN(2); fi; 2*f(n-1)+f(n-2); end;

MATHEMATICA

a=2; b=3; lst={-1, a, b}; Do[c=(a+b)+b; AppendTo[lst, c]; a=b; b=c, {n, 3*4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 13 2009 *)

Table[4 Fibonacci[n, 2] - Fibonacci[n + 1, 2], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 27 2016 *)

PROG

(Haskell)

a078343 n = a078343_list !! n

a078343_list = -1 : 2 : zipWith (+)

                        (map (* 2) $ tail a078343_list) a078343_list

-- Reinhard Zumkeller, Jan 04 2013

(PARI) a(n)=([0, 1; 1, 2]^n*[-1; 2])[1, 1] \\ Charles R Greathouse IV, Jun 11 2015

CROSSREFS

Cf. A000129, A221172, A221173, A221174, A221175.

Sequence in context: A166302 A100342 A041281 * A148038 A148039 A148040

Adjacent sequences:  A078340 A078341 A078342 * A078344 A078345 A078346

KEYWORD

sign,easy

AUTHOR

Benoit Cloitre, Nov 22 2002

EXTENSIONS

Entry revised by N. J. A. Sloane, Apr 29 2004

STATUS

approved

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Last modified September 22 09:19 EDT 2017. Contains 292336 sequences.