A078343 a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).

-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, 411856525348

Offset: 0

Author

Benoit Cloitre, Nov 22 2002

Keywords

• sign
• easy

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

Examples

			G.f. = -1 + 2*x + 3*x^2 + 8*x^3 + 19*x^4 + 46*x^5 + 111*x^6 + ... - _Michael Somos_, Jun 30 2022
		

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
• H. S. M. Coxeter, Numerical distances among the spheres in a loxodromic sequence, Math. Intell. 19(4) 1997 pp. 41-47. See page 41. See pp. 46-47.
• 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:1212.1368 [cs.DM], 2012.
• Index entries for linear recurrences with constant coefficients, signature (2,1).

Crossrefs

Cf. A000129, A048654, A048655, A048746, A111566, A162255, A221172, A221173, A221174, A221175.

Programs

Haskell

a078343 n = a078343_list !! n
a078343_list = -1 : 2 : zipWith (+)
                        (map (* 2) $ tail a078343_list) a078343_list
-- Reinhard Zumkeller, Jan 04 2013

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x)/(-1+2*x+x^2))); // G. C. Greubel, Jul 26 2018

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

Table[4 Fibonacci[n, 2] - Fibonacci[n + 1, 2], {n, 0, 30}] (* Vladimir Reshetnikov, Sep 27 2016 *)
LinearRecurrence[{2,1},{-1,2},40] (* Harvey P. Dale, Apr 15 2019 *)

PARI

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

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)*(-2+3*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

a(n) = -(-1)^n * A048654(-n) = ( (-2+3*sqrt(2))*(1+sqrt(2))^n + (-2-3*sqrt(2))*(1-sqrt(2))^n )/4 for all n in Z. - Michael Somos, Jun 30 2022

2*a(n+1)^2 = A048655(n)^2 + (-1)^n*7. - Philippe Deléham, Mar 07 2023

E.g.f.: 3*exp(x)*sinh(sqrt(2)*x)/sqrt(2) - exp(x)*cosh(sqrt(2)*x). - Stefano Spezia, May 26 2024

Extensions

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