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%I #67 Mar 07 2023 02:43:58
%S -1,2,3,8,19,46,111,268,647,1562,3771,9104,21979,53062,128103,309268,
%T 746639,1802546,4351731,10506008,25363747,61233502,147830751,
%U 356895004,861620759,2080136522,5021893803,12123924128,29269742059,70663408246,170596558551
%N a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2).
%C 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
%D H. S. M. Coxeter, 1998, Numerical distances among the circles in a loxodromic sequence, Nieuw Arch. Wisk, 16, pp. 1-9.
%H Reinhard Zumkeller, <a href="/A078343/b078343.txt">Table of n, a(n) for n = 0..1000</a>
%H H. S. M. Coxeter, <a href="https://doi.org/10.1007/BF03024413">Numerical distances among the spheres in a loxodromic sequence</a>, Math. Intell. 19(4) 1997 pp. 41-47. See page 41. See pp. 46-47.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, <a href="http://arxiv.org/abs/1212.1368">A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake</a>, arXiv:1212.1368 [cs.DM], 2012
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,1).
%F For the unsigned version: a(1)=1; a(2)=2; a(n) = Sum_{k=2..n-1} (a(k) + a(k-1)).
%F a(n) is asymptotic to (1/4)*(-2+3*sqrt(2))*(1+sqrt(2))^n.
%F a(n) = A048746(n-3) + 2, for n>2. - _Ralf Stephan_, Oct 17 2003
%F 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
%F O.g.f.: (1-4*x)/(-1 + 2*x + x^2). - _R. J. Mathar_, Feb 15 2008
%F a(n) = 4*Pell(n) - Pell(n+1), where Pell = A000129. - _Vladimir Reshetnikov_, Sep 27 2016
%F 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
%F 2*a(n+1)^2 = A048655(n)^2 + (-1)^n*7 . - _Philippe Deléham_, Mar 07 2023
%e 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
%p 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;
%t Table[4 Fibonacci[n, 2] - Fibonacci[n + 1, 2], {n, 0, 30}] (* _Vladimir Reshetnikov_, Sep 27 2016 *)
%t LinearRecurrence[{2,1},{-1,2},40] (* _Harvey P. Dale_, Apr 15 2019 *)
%o (Haskell)
%o a078343 n = a078343_list !! n
%o a078343_list = -1 : 2 : zipWith (+)
%o (map (* 2) $ tail a078343_list) a078343_list
%o -- _Reinhard Zumkeller_, Jan 04 2013
%o (PARI) a(n)=([0,1;1,2]^n*[-1;2])[1,1] \\ _Charles R Greathouse IV_, Jun 11 2015
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-4*x)/(-1+2*x+x^2))); // _G. C. Greubel_, Jul 26 2018
%Y Cf. A000129, A048654, A048655, A221172, A221173, A221174, A221175.
%K sign,easy
%O 0,2
%A _Benoit Cloitre_, Nov 22 2002
%E Entry revised by _N. J. A. Sloane_, Apr 29 2004
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